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Limits Calculator

Get detailed solutions to your math problems with our Limits step-by-step calculator. Practice your math skills and learn step by step with our math solver. Check out all of our online calculators here.

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Example

Solved Problems

Difficult Problems

1

Here, we show you a step-by-step solved example of limits. This solution was automatically generated by our smart calculator:

$\lim_{x\to7}\left(\frac{2-\sqrt{x-3}}{x^2-49}\right)$

Plug in the value $7$ into the limit

$\frac{2-\sqrt{7-3}}{7^2-49}$

Subtract the values $7$ and $-3$

$\frac{2-\sqrt{4}}{7^2-49}$

Calculate the power $\sqrt{4}$

$\frac{2-1\cdot 2}{7^2-49}$

Multiply $-1$ times $2$

$\frac{2-2}{7^2-49}$

Subtract the values $2$ and $-2$

$\frac{0}{7^2-49}$

Calculate the power $7^2$

$\frac{0}{49-49}$

Subtract the values $49$ and $-49$

$\frac{0}{0}$
2

If we directly evaluate the limit $\lim_{x\to 7}\left(\frac{2-\sqrt{x-3}}{x^2-49}\right)$ as $x$ tends to $7$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
3

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 7}\left(\frac{\frac{d}{dx}\left(2-\sqrt{x-3}\right)}{\frac{d}{dx}\left(x^2-49\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(2-\sqrt{x-3}\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(2\right)+\frac{d}{dx}\left(-\sqrt{x-3}\right)$

The derivative of the constant function ($2$) is equal to zero

$\frac{d}{dx}\left(-\sqrt{x-3}\right)$

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$-\frac{d}{dx}\left(\sqrt{x-3}\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$- \left(\frac{1}{2}\right)\left(x-3\right)^{-\frac{1}{2}}\frac{d}{dx}\left(x-3\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$- \left(\frac{1}{2}\right)\left(x-3\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-3\right)\right)$

Multiplying the fraction by $-1$

$-\frac{1}{2}\left(x-3\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-3\right)\right)$

The derivative of the constant function ($-3$) is equal to zero

$-\frac{1}{2}\left(x-3\right)^{-\frac{1}{2}}\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$-\frac{1}{2}\left(x-3\right)^{-\frac{1}{2}}$

Find the derivative of the denominator

$\frac{d}{dx}\left(x^2-49\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-49\right)$

The derivative of the constant function ($-49$) is equal to zero

$\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$

Divide fractions $\frac{-\frac{1}{2}\left(x-3\right)^{-\frac{1}{2}}}{2x}$ with Keep, Change, Flip: $\frac{a}{b}\div c=\frac{a}{b}\div\frac{c}{1}=\frac{a}{b}\times\frac{1}{c}=\frac{a}{b\cdot c}$

$\lim_{x\to7}\left(\frac{-\left(x-3\right)^{-\frac{1}{2}}}{4x}\right)$
4

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to7}\left(\frac{-\left(x-3\right)^{-\frac{1}{2}}}{4x}\right)$
5

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$\lim_{x\to7}\left(\frac{-1}{4x\sqrt{x-3}}\right)$
6

Evaluate the limit $\lim_{x\to7}\left(\frac{-1}{4x\sqrt{x-3}}\right)$ by replacing all occurrences of $x$ by $7$

$\frac{-1}{4\cdot 7\sqrt{7-3}}$
7

Subtract the values $7$ and $-3$

$\frac{-1}{4\cdot 7\sqrt{4}}$
8

Multiply $4$ times $7$

$\frac{-1}{28\sqrt{4}}$
9

Calculate the power $\sqrt{4}$

$\frac{-1}{28\cdot 2}$
10

Multiply $28$ times $2$

$-\frac{1}{56}$

Final answer to the problem

$-\frac{1}{56}$

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