Final Answer
$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$
Step-by-step explanation
Problem to solve:
$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$
Choose the solving method
1
Applying rationalisation
$\lim_{x\to10}\left(\frac{\sqrt{x+6}-4}{x-10}\frac{\sqrt{x+6}+4}{\sqrt{x+6}+4}\right)$
2
Multiplying fractions $\frac{\sqrt{x+6}-4}{x-10} \times \frac{\sqrt{x+6}+4}{\sqrt{x+6}+4}$
$\lim_{x\to10}\left(\frac{\left(\sqrt{x+6}-4\right)\left(\sqrt{x+6}+4\right)}{\left(x-10\right)\left(\sqrt{x+6}+4\right)}\right)$
3
Solve the product of difference of squares $\left(\sqrt{x+6}-4\right)\left(\sqrt{x+6}+4\right)$
$\lim_{x\to10}\left(\frac{-10+x}{\left(x-10\right)\left(\sqrt{x+6}+4\right)}\right)$
4
Simplify the fraction $\frac{-10+x}{\left(x-10\right)\left(\sqrt{x+6}+4\right)}$ by $-10+x$
$\lim_{x\to10}\left(\frac{1}{\sqrt{x+6}+4}\right)$
5
Evaluate the limit $\lim_{x\to10}\left(\frac{1}{\sqrt{x+6}+4}\right)$ by replacing all occurrences of $x$ by $10$
$\frac{1}{\sqrt{10+6}+4}$
6
Simplifying, we get
$\frac{1}{8}$
Final Answer
$\frac{1}{8}$$\,\,\left(\approx 0.125\right)$