Step-by-step Solution

Evaluate the limit of $\frac{\sqrt{x+6}-4}{x-10}$ as $x$ approaches $10$

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Final Answer

$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$

Step-by-step explanation

Problem to solve:

$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$

Choose the solving method

1

If we try to evaluate the limit directly, it results in indeterminate form. Then we need to apply L'Hôpital's rule

$\lim_{x\to10}\left(\frac{\frac{d}{dx}\left(\sqrt{x+6}-4\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
2

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

$\lim_{x\to10}\left(\frac{\frac{d}{dx}\left(\sqrt{x+6}\right)+\frac{d}{dx}\left(-4\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
3

The derivative of the constant function ($-4$) is equal to zero

$\lim_{x\to10}\left(\frac{\frac{d}{dx}\left(\sqrt{x+6}\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
4

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

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}\frac{d}{dx}\left(x+6\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
5

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

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(6\right)\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
6

The derivative of the constant function ($6$) is equal to zero

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}\frac{d}{dx}\left(x\right)}{\frac{d}{dx}\left(x-10\right)}\right)$
7

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

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}}{\frac{d}{dx}\left(x-10\right)}\right)$
8

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

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}}{\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-10\right)}\right)$
9

The derivative of the constant function ($-10$) is equal to zero

$\lim_{x\to10}\left(\frac{\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}}{\frac{d}{dx}\left(x\right)}\right)$
10

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

$\lim_{x\to10}\left(\frac{1}{2}\left(x+6\right)^{-\frac{1}{2}}\right)$
11

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

$\frac{1}{2}\lim_{x\to10}\left(\left(x+6\right)^{-\frac{1}{2}}\right)$
12

Evaluate the limit by replacing all occurrences of $x$ by $10$

$\frac{1}{2}\cdot \left(10+6\right)^{-0.5}$
13

Simplifying

$\frac{1}{2}\cdot \frac{1}{4}$
14

Multiply $\frac{1}{2}$ times $\frac{1}{4}$

$\frac{1}{8}$

Final Answer

$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$
$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$

Main topic:

Limits

Related formulas:

4. See formulas

Steps:

14

Time to solve it:

~ 0.06 s (SnapXam)