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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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###  Difficult Problems

1

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

$\lim_{x\to4}\left(\frac{x^2-16}{x^2+2x-24}\right)$
2

Factor the trinomial $x^2+2x-24$ finding two numbers that multiply to form $-24$ and added form $2$

$\begin{matrix}\left(-4\right)\left(6\right)=-24\\ \left(-4\right)+\left(6\right)=2\end{matrix}$
3

Rewrite the polynomial as the product of two binomials consisting of the sum of the variable and the found values

$\lim_{x\to4}\left(\frac{x^2-16}{\left(x-4\right)\left(x+6\right)}\right)$

Plug in the value $4$ into the limit

$\frac{4^2-16}{\left(4-4\right)\left(4+6\right)}$

Calculate the power $4^2$

$\frac{16-16}{\left(4-4\right)\left(4+6\right)}$

Subtract the values $16$ and $-16$

$\frac{0}{\left(4-4\right)\left(4+6\right)}$

Subtract the values $4$ and $-4$

$\frac{0}{0\left(4+6\right)}$

Add the values $4$ and $6$

$\frac{0}{0\cdot 10}$

Multiply $0$ times $10$

$\frac{0}{0}$
4

If we directly evaluate the limit $\lim_{x\to 4}\left(\frac{x^2-16}{\left(x-4\right)\left(x+6\right)}\right)$ as $x$ tends to $4$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
5

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

$\lim_{x\to 4}\left(\frac{\frac{d}{dx}\left(x^2-16\right)}{\frac{d}{dx}\left(\left(x-4\right)\left(x+6\right)\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(x^2-16\right)$

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

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

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

$2x$

Find the derivative of the denominator

$\frac{d}{dx}\left(\left(x-4\right)\left(x+6\right)\right)$

Multiply the single term $x+6$ by each term of the polynomial $\left(x-4\right)$

$\frac{d}{dx}\left(x\left(x+6\right)-4\left(x+6\right)\right)$

Multiply the single term $x$ by each term of the polynomial $\left(x+6\right)$

$\frac{d}{dx}\left(x\cdot x+6x-4\left(x+6\right)\right)$

When multiplying two powers that have the same base ($x$), you can add the exponents

$\frac{d}{dx}\left(x^2+6x-4\left(x+6\right)\right)$

Multiply the single term $-4$ by each term of the polynomial $\left(x+6\right)$

$\frac{d}{dx}\left(x^2+6x-4x-4\cdot 6\right)$

Multiply $-4$ times $6$

$\frac{d}{dx}\left(x^2+6x-4x-24\right)$

Combining like terms $6x$ and $-4x$

$\frac{d}{dx}\left(x^2+2x-24\right)$

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

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

The derivative of the linear function times a constant, is equal to the constant

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

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

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

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

$2x+2$

Factor the denominator by $2$

$\lim_{x\to4}\left(\frac{2x}{2\left(x+1\right)}\right)$

Cancel the fraction's common factor $2$

$\lim_{x\to4}\left(\frac{x}{x+1}\right)$
6

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

$\lim_{x\to4}\left(\frac{x}{x+1}\right)$

Evaluate the limit $\lim_{x\to4}\left(\frac{x}{x+1}\right)$ by replacing all occurrences of $x$ by $4$

$\frac{4}{4+1}$

Add the values $4$ and $1$

$\frac{4}{5}$
7

Evaluate the limit $\lim_{x\to4}\left(\frac{x}{x+1}\right)$ by replacing all occurrences of $x$ by $4$

$\frac{4}{5}$

##  Final answer to the problem

$\frac{4}{5}$

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