$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$
$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$
Limits by L'Hôpital's rule
Limits by factoring
Limits by rationalizing
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}$
$\frac{1}{2}\cdot \frac{1}{4}$
14
Multiply $\frac{1}{2}$ times $\frac{1}{4}$
$\frac{1}{8}$
$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$