Category: Education

In mathematics, limits are used to solve the complex calculus problems of various functions. It is mainly used to define differential, continuity, and integrals. Limits accomplished a particular value function by substituting the limit value.

Limits are very essential in a type of antiderivative known as definite integral in which upper and lower limits are applied. In this post, we’ll learn the definition and rules of limits with a lot of examples.

What are the limits in calculus?

In calculus, a value that a function approaches as an input of that function gets closer and closer to some specific number is known as limit. In other words, when a function approaches to some value to evaluate the value of limit of that function is known as limits.

To measure the nearness and representation of mathematical concept ideas, the limit’s notation can be used. It is very beneficial for defining other branches of calculus like derivative, continuity, and antiderivative.

The equation of limits

The formula or equation used to calculate the limits of the functions is given below.

lim_x→u h(x) = N

• Lim is the notation of limits.
• u is the limit value of the function.
• h(x) is the given function.
• x is the variable of the function.
• N is the result of the function after applying the limit value u.

You have to apply the limit value u to the given function h(x), for solving the problems of limits. The limits are not applied on the constant functions so the limits of constant functions remain unchanged.

Rules of limits

There are various rules of limits in calculus. Let’s discuss them briefly with the help of examples to evaluate the limit problems.

1.   Constant rule

According to this rule of limits, the constant function remains the same. Because limits are applied only on the variables. The equation for the constant rule is:

lim_x→u C = C

• Lim is the notation of limits.
• u is the limit value of the function.
• C is the given function.

Example

Find the limit of 56 as x approaches to 5.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→5 56

Step 2: Now apply the limit.

lim_x→5 56 = 56 (by constant rule)

2.   Constant function rule

According to this rule of limits, the constant with the function will be written outside the limit notation. Because limits are applied only on the variables. The equation for the constant function rule is:

lim_x→u C h(x) = C * N

• Lim is the notation of limits.
• u is the limit value of the function.
• C is any constant.
• h(x) is the given function.
• x is the variable of the function.
• N is the result of the function after applying the limit value u.

Example

Find the limit of 23x³ as x approaches to 7.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→7 23x³

Step 2: Now apply the constant function rule of limit.

lim_x→7 23x³ = 23 lim_x→7 x³

Step 3: Now apply the limit.

lim_x→7 23x³ = 23 (7³)

lim_x→7 23x³ = 23 (7 * 7 * 7)

lim_x→7 23x³ = 23 (343)

lim_x→7 23x³ = 7889

3.   Sum rule

According to this rule of limits, the notation applied to each function separately. The equation for the sum rule is:

lim_x→u [h(x) + g(x)] = lim_x→u (h(x)) + lim_x→u (g(x)) = M + N

• Lim is the notation of limits.
• u is the limit value of the function.
• g(x) & h(x) are the given functions.
• x is the variable of the function.
• M & N are the results of the functions after applying the limit value u.

Example

Find the limits of x³ + x⁵ as x approaches to 3.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→3 [x³ + x⁵]

Step 2: Now apply the sum rule of limit.

lim_x→3 [x³ + x⁵] = lim_x→3 (x³) +  (x⁵)

Step 3: Now apply the limit.

lim_x→3 [x³ + x⁵] = (3³) + (3⁵)

lim_x→3 [x³ + x⁵] = (3 * 3 * 3) + (3 * 3 * 3 * 3 * 3)

lim_x→3 [x³ + x⁵] = (27) + (243)

lim_x→3 [x³ + x⁵] = 270

4.   Difference rule

According to this rule of limits, the notation applied to each function separately. The equation for the difference rule is:

lim_x→u [h(x) – g(x)] = lim_x→u (h(x)) – lim_x→u (g(x)) = M – N

• Lim is the notation of limits.
• u is the limit value of the function.
• g(x) & h(x) are the given functions.
• x is the variable of the function.
• M & N are the results of the functions after applying the limit value u.

Example

Find the limits of x³ – x⁵ as x approaches to 2.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→2 [x³ – x⁵]

Step 2: Now apply the difference rule of limit.

lim_x→2 [x³ – x⁵] = lim_x→2 (x³) + lim_x→2 (x⁵)

Step 3: Now apply the limit.

lim_x→2 [x³ – x⁵] = (2³) – (2⁵)

lim_x→2 [x³ – x⁵] = (2 * 2 * 2) – (2 * 2 * 2 * 2 * 2)

lim_x→2 [x³ – x⁵] = (8) – (32)

lim_x→2 [x³ – x⁵] = -24

5.   Product rule

According to this rule of limits, the notation applied to each function separately. The equation for the product rule is:

lim_x→u [h(x) * g(x)] = lim_x→u (h(x)) * lim_x→u (g(x)) = M * N

• Lim is the notation of limits.
• u is the limit value of the function.
• g(x) & h(x) are the given functions.
• x is the variable of the function.
• M & N are the results of the functions after applying the limit value u.

Example

Find the limits of x⁵ * x³ as x approaches to 4.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→4 [x⁵ * x³]

Step 2: Now apply the product rule of limit.

lim_x→4 [x⁵ * x³] = lim_x→4 (x⁵) * lim_x→4 (x³)

Step 3: Now apply the limit.

lim_x→4 [x⁵ * x³] = (4⁵) * (4³)

lim_x→4 [x⁵ * x³] = (4 * 4 * 4 * 4 * 4) * (4 * 4 * 4)

lim_x→4 [x⁵ * x³] = (1024) * (64)

lim_x→4 [x⁵ * x³] = 65536

6.   L’hospital rule

According to this rule, if the function forms 0 by 0 or infinity by infinity form after applying the limits then take the derivatives of the numerator and the denominator and then apply the limit value again.

Example

Find the limit of x² – 4 / 4x – 2x² as x approaches to 2.

Solution

Step 1: Apply the limit notation on the given function.

lim_x→2 [x² – 4 / 4x – 2x²]

Step 2: Now apply the quotient rule of limit and apply the limit value.

lim_x→2 [x² – 4 / 4x – 2x²] = lim_x→2 [x² – 4] / lim_x→2 [4x – 2x²]

lim_x→2 [x² – 4 / 4x – 2x²] = [2² – 4] / [4(2) – 2(2)²]

lim_x→2 [x² – 4 / 4x – 2x²] = [4 – 4] / [4(2) – 2(4)]

lim_x→2 [x² – 4 / 4x – 2x²] = [4 – 4] / [8 – 8]

lim_x→2 [x² – 4 / 4x – 2x²] = 0 / 0

Step 3: As the functions make zero by zero form by applying the limit value, so apply the L’hospital rule and apply the limit value again.

lim_x→2 [x² – 4 / 4x – 2x²] = lim_x→2 [d/dx (x² – 4) / d/dx (4x – 2x²)]

lim_x→2 [x² – 4 / 4x – 2x²] = lim_x→2 [2x – 0 / (4 – 4x)]

lim_x→2 [x² – 4 / 4x – 2x²] = lim_x→2 [2x / (4 – 4x)]

lim_x→2 [x² – 4 / 4x – 2x²] = [2(2) / (4 – 4(2))]

lim_x→2 [x² – 4 / 4x – 2x²] = [4 / (4 – 8)]

lim_x→2 [x² – 4 / 4x – 2x²] = [4 / (– 4)]

lim_x→2 [x² – 4 / 4x – 2x²] = -1

Summary

In this post, we have learned the definition and rules of limits with a lot of examples. Now you can grab all the basics of limits just by following the rules of this post.

Explore These Additional Math Resources (Parents, Teachers and Students)

• The Importance of Methodology to Make Learning Mathematics Easier
• Why Do Students Find Limit Calculation Difficult
• How to Recognize and Overcome Math Anxiety

Coding is a subject and field of study that most kids across the globe get recommended in their curriculum. Research on the skills and mindset of a child suggests and reveals that programming languages have immense benefits for them. It can help develop their cognitive, problem-solving, and decision-making abilities.

In addition, it aids in enhancing their persistence and tenacity.  However, coding for kids can involve some obstacles. These problems may hinder their progress or even decrease their interest in the worst cases.

In this article, let us discuss the most significant issues a child may face when learning to code.

Increased Focus on Coding Syntax

Coding involves the implementation of an algorithm into a code. It can get done with the help of a programming language. Thus, the system and process work using an appropriate syntax. It depends on the chosen programming language.

The syntax ensures the smooth running of a code. Nevertheless, the essence of the subject lies in problem-solving skills. The abilities show their color when a running or syntax error arises. However, most courses, students, and even teachers do not pay much attention to this essential skill. Instead, they focus more on the syntax.

Lack of Live Help

It is possible to learn coding from various online courses. On top of that, the compilers and debuggers help detect and rectify the errors that arise along the way. However, learning in this process proves exceedingly taxing and time-consuming. In addition, a kid may lose interest in the subject because of the multiple failures they face.

For that reason, it is necessary to get the guidance of a reliable and experienced teacher when learning to code. It helps sustain a kid’s interest and engagement in the subject. However, it is arduous nowadays to find a teacher who remains willing to put in their utmost effort. Moreover, the events of recent months has dissuaded the instructors from teaching in real-time.

Less Practice

The skill and knowledge of coding revolve around extensive practice. Nevertheless, most online courses or offline instructors focus on the teaching part. They omit the practice section and time entirely. It proves detrimental to the development and growth of the student. On top of that, some kids may not get the platforms required to run their programs and codes. In such cases as well, they face an issue. It can get owed to the lack of hands-on experience and visual perception of the running process and output.

Low Motivation and Engagement

One of the most significant obstacles for kids when learning to code is low motivation, dedication, and engagement. In most cases, they remain disinterested in the subject and fail to develop an attraction and appeal no matter how much they get involved with it. It may be the scenario even if the course or instructor allows for maximum comfort, convenience, or effortlessness in the learning process and journey.

Tendency to Forget

It is not uncommon for someone to forget what they have learned about something. Kids learning to code is no exception. They may acquire knowledge about the subject and segregate it into their short-term memory. It implies that they cannot retain the ideas and concepts for long. Such an issue gets enhanced by the lack of practice.  

How Can a Kid Overcome Obstacles When Learning Coding?

Issues and obstacles are bound to appear when learning something new. However, what matters are the ways and methods to overcome them. It implies that even kids can do that.

For instance, a child can strengthen their memory if they tend to forget things fast. Additionally, they can overcome their lack of practice on a computer platform by writing the programs and code by hand. Initially, the hand-written material would help to build their fundamentals. Then, when they get a required platform, they can run them there to check.

One of the benefits of coding for kids is increased focus as the engage in learning.  This can help overcome a lack of motivation by incorporating fun and exciting codes and outputs. It would make kids look forward to completing their work. Kids can develop their problem-solving skills by playing various games like chess, cross the bridge, etc. Lastly, it is possible to seek live help or instructors by searching the internet or through acquaintances.

7 Things To Consider Before Joining Online Coding Classes For Kids

Most schools have now reopened and many children are reaping the social and educational benefits of in-person learning.  However, if your child thrived through online learning or is immunocompromised, it might be a good idea to keep them enrolled in a remote-learning school.

Additionally, school closures for things like snow days are now a thing of the past — teachers and students alike are well set up to offer online learning in lieu of physical classrooms.

If you’re a parent to a school-aged child, you simply must know how to help your child focus during online learning. Here are a few of the best practices to consider.

A Dedicated Space

Children may have a hard time separating their educational career from their personal life when learning at home. This is understandable, as we would all struggle to focus if our favorite toys or TV programs were just a few feet out of view. You can help children avoid distractions by creating a dedicated space that facilitates learning and engagement.

The space you choose for an at-home classroom will largely be dependent on the layout of your home. In general, though, your child will do best in a room where you can observe them, yet they feel as though they have privacy and are able to contribute to class discussions without worrying about being too loud.

The space you choose should also be completely free of clutter. Clutter is a major distraction for kids during classes they don’t enjoy, and can even pose a health hazard to younger children. You can declutter your home by intentionally organizing everything you own, and giving everything its own place. You should also try giving your child a home-desk, where they can store all their learning materials.

Fidgeting

Fidgeting is entirely normal behavior for children and adults alike. Some research even suggests that fidgeting and fidget toys can help children relax and focus during classes. However, as a parent, you probably have a good idea of when your child is fidgeting due to boredom or to distract themselves.

You can embrace productive fidgeting by providing your child with the kind of fidget toy or stimulus that suits their kind of fidgeting. For example, if they are body fidgeters, you can give them an exercise ball to sit on for part of the day, Alternatively, if you find that your child has a busy mouth, and likes to click or blow bubbles when they should be learning, you can give them caffeine-free tea to keep them occupied and well hydrated throughout the day.

Focus Throughout the Day

The hardest part about school is that children are expected to retain a high level of focus throughout the day. Unlike us adults, who are able to front-load more difficult work in the morning when we have more focus, children have to complete complex tasks in classes like math or literature regardless of the time of day or how they feel.

You can help your child retain their focus throughout the day by giving them brain breaks between classes or during their lunchtime. Ideally, these brain breaks will not involve screens and will get them up and moving. For example, a walk around the block, or even allowing them to help cook lunch will take their mind off learning, and help them return to the classroom with greater focus.

You also need to be aware of the food your child is eating, and how much water they are drinking. Typically, a school day diet should consist of whole foods with antioxidant-rich berries, protein sources, leafy greens, and plenty of water to keep them hydrated.

Testing for ADHD

Sometimes seeking help is the best thing you can do for your child. It doesn’t take much to have your child tested for ADHD, and the stigma around mental health conditions is finally being challenged and reversed in most areas of society.

According to the CDC, the symptoms of ADHD include inattention (difficulty organizing tasks, failure to pay close attention during classroom activities, forgetfulness during daily activities, etc.) and hyperactivity or impulsivity (fidgets or squirms in seat, talks excessively, interrupts conversations frequently).

The diagnosis of ADHD is typically made by your primary healthcare provider, who can then offer a range of treatment options that usually involve either behavioral and environmental changes or medication.

As a parent, it’s important to recognize that testing your child for ADHD does not mean you’ve “failed” them. You’re doing the right thing by reaching out to professional medical advice, and are showing your child that it’s ok to seek help when they need it.

Conclusion

Online learning can be great for children who struggle in traditional classrooms or aren’t able to return to a physical school environment.

However, maintaining a high level of focus throughout the day can be difficult — particularly if your home environment isn’t conducive to learning. You can help your child focus during remote classes by removing distractions from their at-home classroom, and by decluttering their space regularly. You should also reconsider the food you give them, and opt for focus-friendly whole foods that help them stay on track and engaged in their learning experience.

About the Author
Katie Brenneman is a passionate writer specializing in education, mental health, family lifestyle and online safety. When she isn’t writing, you can find her with her nose buried in a book or hiking with her dog, Charlie. You can follow her on Twitter.

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Teachers:  Discover  online class productivity tips to help keep kids engaged.

Your child has been accepted for college and you couldn’t be prouder. However, if this is the first time you’ve had to wave a child off to college, there are probably many things going around in your head about how they are going to cope.  Apprehension, mixed with excitement for your teen, is natural for any parent.

Here are our top five tips for parents of new college students to help you and your child through the first year.

Encourage Your Child to Keep in Close Contact with Their Academic Advisor

Regularly checking in with their course advisor will help keep your child on track with their studies. It will also ensure that they don’t end up in a situation where they must take extra classes (at an extra cost) to catch up. Their advisor will help keep them on track with their credits and ensure they’re making the required progress towards their degree.

It’s always easier to nip any problems in the bud than to try and catch up later. These professionals can also help with things like creating a recommendation letter, finding clubs on campus, or offering valuable study tips so building and maintain a relationship with them is a great idea.

Investigate Local Scholarships or Those Offered by the University

Set up a meeting for you and your child with the university’s scholarship department to learn more about available scholarships. Quite often the university will hold general meetings to provide details of funding offered. Encourage your child to apply for scholarships and grants online. They can build a profile before they can be matched to eligible scholarships for college. If something suitable is available, they can then apply directly.

Note that funds from a scholarship can be added to the student finance account at any time during the school year, not just at financial check-in. So, if you haven’t already looked at scholarships, it’s not too late.

Keep Records of All College Expenses That Are Tax Deductible

If you are supporting your child financially, the college will send you a tax form at the end of the year. Ask your child to keep and send you receipts for purchases that are related to their coursework such as books, supplies and equipment. You can then use these as tax deductibles.

Find Out if There is a Parent Program at the College

We all know how our children forget to keep us informed about what’s going on. Joining a parent program will help to keep you in the loop about any opportunities and events for parents and families on campus. The program may include activities such as family sports days, barbecues, and tailgate parties. Ask about the possibility of joining at move-in or orientation day and sign up for any newsletters that may be available.

Find Out What Books They Will Actually Need

Don’t rush out and buy all the books on the reading list. You may find yourself paying for books that are never used. Most textbooks can be found at used book sites online or can be loaned from a library. If a student emails the teacher in advance, they can ask if a particular textbook is needed. Sometimes they are not. Students can also access books via online forums from prior students of the course either for free or a very low cost.

Nurture Independence

As a parent, your instinct can be to help your kid as much as possible. The problem with this well-meaning impulse is that it can result in a child growing toward adulthood who is unprepared to solve problems on their own. It can feel good when your kid needs you, but it’s to their detriment if you prize this over letting them be independent. It’s important to let them feel proportionate consequences for their actions; if they don’t study for a test or complete an assignment, they won’t get a good grade on it.

Dealing with the Change in Your Life

The academic life of a child is a time of nearly constant change.  It can be often be tough to cope with, especially if they no longer want to do the same activities with you that they used to or seem to be pulling away from you. It’s important to remind yourself that as long as they aren’t moving toward something harmful, this pulling away is a sign you’ve been successful. Of course, you can continue to be a support to your child while preparing them to be their own person and to take on the world with confidence.

Resources for University and College Prep

• Should your child go to trade school or college?