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

Applying rationalisation

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

Multiplying fractions $\frac{\sqrt{x+6}-4}{x-10} \times \frac{\sqrt{x+6}+4}{\sqrt{x+6}+4}$

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

Solve the product of difference of squares $\left(\sqrt{x+6}-4\right)\left(\sqrt{x+6}+4\right)$

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

Simplify the fraction $\frac{-10+x}{\left(x-10\right)\left(\sqrt{x+6}+4\right)}$ by $-10+x$

$\lim_{x\to10}\left(\frac{1}{\sqrt{x+6}+4}\right)$
5

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

$\frac{1}{\sqrt{10+6}+4}$
6

Simplifying

$\frac{1}{8}$

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

Steps:

6

Time to solve it:

~ 0.04 s (SnapXam)