Limit Calculator

Limit Calculator – Step by Step Guide to Solving Limits Online

What if someone tells you that the fastest runner in Greece could not overtake a slow moving tortoise? Sounds silly, right? But this is what Zeno’s paradox, a 2000 year old tale claimed. In this experiment, Achillies gives a tortoise a head-start in a race. By the time he reaches the tortoise’s starting point, the tortoise has moved ahead. Achillies moves further ahead to reach the new spot but till that time, the tortoise moves further ahead. This happens indefinitely. Strangely, Achillies never catches up. So how can we make sense of this endlessly seeming chase?

This idea of approaching something without actually reaching it, is the heart of a concept in Mathematics called limits. Limits help us acknowledge the value of a function, not particularly at a specific input number, but at what approaches the number. It is a powerful and evidently great tool to calculate the value of a function where direct substitution is not possible like dividing any number by zero.

Limits form the base of many different and important branches of Mathematics such as integrals, derivatives and continuity. Just like we can understand motion by observing at how objects approach each other in the real world, in Mathematics, limits help us analyze how functions behave as their variables get close to a certain point even when the value isn’t directly defined there.

Origin of Limits

Actually, limits have been originated from the need to understand motion, change and value of functions where they are not actually defined or are incomprehensible. Although it was not until 19th century that a formal and precise definition of limits, using the epsilon-delta method, was provided by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass. This understanding was one of the biggest milestones in history in making Mathematics as reliable as it is today.

What is a Limit?

The value of a function f(x) as x approaches a number, a is called the function’s limit. x does not actually take the value of a, it just approaches it and is infinitely close to c. This can be written as:

$\lim _{x\to a}\left(f\left(x\right)\right)=L$ This means that as x approaches a, f(x) approaches L. Now, anything can be approached either from its left or its right side, thus, we have RHL and LHL.

LHL (Left Hand Limit) : When a function is approached from its left or negative side $(x\to a^-)$, it is called its Left Hand Limit.

RHL (Right Hand Limit) : When a function is approached from its right or positive side $(x\to a^+)$, it is called its Right Hand Limit. If RHL equals LHL, then we can say that the overall limit exists.

Types of Limits

Limits can be categorized into 3 types : finite limits, infinite limits and one-sided limits. Let us look deeply into these.

Finite Limits : A finite limit is one where when x approaches a value, it gives out a finite, defined or finite output.

Example:

Given : f(x) = 2x + 1

Then,

$\lim _{x\to 3}\left(2x+1\right)=7$

Here, as x approaches 3, the function f(x) gets closer to 7, which is a finite number.

Infinite Limits: Here, the output approaches an undefined value(positive infinity or negative infinity) as the value of x approaches a certain number.

Example:

Given : f(x) = 1x

Now, as x approaches 0 from the right hand side, the function approaches positive infinity.

$\lim _{x\to 0^+}\frac{1}{x}=+\infty$

Similarly, as x approaches 0 from the left hand side, the function approaches negative infinity.

$ \lim _{x\to 0^-}\frac{1}{x}=-\infty$

One sided Limits : A one-sided limit looks at what the function is approaching from only one side, either from the left side or the right side of a point.

Example :

Given :

$ f\left(x\right)=1$ if $x<2$ and $f\left(x\right)=3$ if $x\ge 2$ Then,

$\lim _{x\to 2^-}\left(f\left(x\right)\right)=1$ (from the left)

$\lim _{x\to 2^+}\left(f\left(x\right)\right)=3$ (from the right)

Since the left and right-hand limits are not equal, the overall limit does not exist at x = 2.

Different methods to solve for limits

There can be different methods to solve a limit depending upon the type of question. Let us discuss these :

Direct substitution method : This is the most direct and easiest method to find a limit and we can use this when the function is continuous at the point the limit is being evaluated.

Steps :

Plug the value of x at which the limit is approaching directly into the function.

If you get valid result, that is the limit. If not, other methods have to be tried.

Example :

$\lim _{x\to 3}\left(2x+1\right)$

Here, if we put x = 3 directly, we get –

2(3) + 1 = 7

7 is a finite value. Thus, 7 is the solution.

Factorization method : This method can be used when by directly putting the value, the output comes out to be an indeterminate form like 00.

Steps :

Factorize both, the numerator and the denominator both.

If they have a common factor, cut it to simplify the expression.

Now, substitute the value of x into the simplifies expression.

Example :

$\lim _{x\to 2}\left(\frac{x^2-4}{x-2}\right)$

Factorize the numerator,

= $x^2-2=\left(x+2\right)\left(x-2\right)$

So,

$\frac{x^2-2}{x-2}=x+2$

Now, put x = 2 in the equation directly.

= 2 + 2 = 4

Thus, the answer is 4.

L'Hopital's Rule : This is typically used when an indeterminate form of $\frac{0}{0}$ is formed. It allows us to differentiate the numerator and denominator individually and re-evaluate the limit.

Steps :

Substitute the given value directly into the function first. If it gives indeterminate form, use L'Hopital's Rule.

Now, differentiate the numerator and denominator individually.

Put the value of x into the differentiated numerator and denominator and get the solution.

Example :

$F\left(x\right)=\lim _{x\to 0}\left(\frac{e^x-1}{x}\right)$

If we substitute x = 0 directly in the equation, it gives $\frac{0}{0}$ indeterminate form. So we will have to differentiate the numerator and denominator individually. So, it becomes : $\lim _{x\to 0}\left(\frac{e^x}{1}\right)$

Now, substitute x with 0. We get :

$\frac{e^0}{1} $

= $\frac{1}{1}$

= 1

Thus, solution to this question is 1

Rationalization Rule : For functions involving square roots, we use rationalization rule if putting the value in the function gives indeterminate value like $\frac{0}{0}$.

Steps :

Look for square roots in numerator or denominator of the function.

To rationalize the square root, multiply the numerator and denominator by the conjugate of the square root.

Simplify the given expression and substitute the given value in the simplified expression.

Example :

$F(x) = \lim _{x\to 0}\left(\frac{\sqrt{x+1}-1}{x}\right)$

Try to substitute x =0 directly in the function. This gives us \frac{0}{:0} indeterminate form. Now that we have come to know that it is in indeterminate form, we can try to rationalize.

Here we will rationalize the numerator by multiplying the numerator and denominator by $\sqrt{x+1}+1$

= $\frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}\times \frac{\sqrt{x+1}-1}{x}$

= $\frac{x}{x\times \left(\sqrt{x+1}+1\right)}$

= $\frac{1}{\left(\sqrt{x+1}+1\right)}$

Now, we no longer have an indeterminate form. So, we can directly put x = 0 in this equation.

= $\frac{1}{\left(\sqrt{0+1}+1\right)}$

= $\frac{1}{2}$

Trigonometric Limits : In calculus, trigonometric limits refer to the limits in which trigonometric expressions like Cos(x), Sin(x) etc. are used and these expressions approach a value when x approaches some given value.

These limits often come to use when direct substitution of values in the given function gives indeterminate forms like $\frac{0}{0} or \frac{\infty }{\infty }$

Key Trigonometric Limits :

$\lim _{x\to 0}\left(\frac{\sin \left(x\right)}{x}\right)=1 $

$\lim _{x\to 0}\left(\frac{tan\left(x\right)}{x}\right)=1$

$\lim _{x\to 0}\left(\frac{1-cos\left(x\right)}{x}\right)=0$

Example :

$f\left(x\right)=\lim _{x\to :0}\left(\frac{\sin \left(3x\right)}{x}\right)$

We can use $\lim _{x\to 0}\left(\frac{\sin \left(x\right)}{x}\right)=$ to solve this question. But first, we need to rewrite this expression in some other manner so that the Trigonometric limit can be used.

Let us take 3x = m

So, $x=\frac{m}{3}$

As $x\rightarrow 0$,$m\rightarrow 0$

So,

$\lim _{x\to 0}\left(\frac{\sin \left(3x\right)}{x}\right) = \lim _{x\to 0}\left(\frac{\sin \left(m\right)}{\frac{m}{3}}\right)$

= $\lim _{x\to 0}3\left(\frac{\sin \left(m\right)}{m}\right)$

Using the trigonometric limit, $\lim _{x\to 0}\left(\frac{\sin \left(m\right)}{m}\right)=1$

So,

$f\left(x\right)=\lim _{x\to 0}\left(\frac{\sin \left(3x\right)}{x}\right)=3\times 1=3$

How to use a Limits Calculator

Enter Your Problem: Type in your equation, expression, or system into the calculator's input field.

Select the operation: Choose the function you need: solve, simplify, factor, graph, etc.

Click Calculate: The calculator processes your input and provides a detailed solution.

Review the Steps: The step-by-step explanation helps you understand the process and learn how to solve similar problems.

Example :

Find $f\left(x\right)=\lim _{x\to 0}\left(\frac{tan\left(3x\right)}{x}\right)$

Calculator solution:

Step 1 : Open the limit calculator

Step 2 : Choose the ‘lim’ option.

$\lim _{x\to \infty }\left(\right)$ should be seen on your screen

Step 3 : Change the limit from approaching infinite to approaching 0 manually

Step 4 : Choose the ‘fraction’ option and put tan(3x) in numerator and x in denominator.

Step 5 : Press ‘Go’ and you can see the step-wise solution there.

Benefits of Using Limits Calculator

1. Saves time and provides accurate solutions.

2. Shows step-by-step solutions for learning.

3. Useful for students and teachers.

4. Online accessibility and free usage.