Solved Problems

Tag: indeterminate form

Problem 2-5: Evaluating Indeterminate Form

Calculate $\displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}$ .

Solution
$\frac{\sqrt{x+2}-2}{x-2}=\frac{0}{0}$
$\displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}= \displaystyle\lim_{x\to 2}\frac{\sqrt{x+2}-2}{x-2}\times \frac{\sqrt{x+2}+2}{\sqrt{x+2}+2}$

$=\displaystyle\lim_{x\to 2}\frac{x+2-4}{(x-2) \times (\sqrt{x+2}+2)}$
$=\displaystyle\lim_{x\to 2}\frac{x-2}{(x-2) \times (\sqrt{x+2}+2)}$

$=\displaystyle\lim_{x\to 2}\frac{1}{(\sqrt{x+2}+2)}=\frac{1}{4}$

April 9, 2010
Problem 2-4: Computing Limits of a Rational Function

Compute
a) $\displaystyle\lim_{x\to 3}\frac{x^2+2x-3}{x+3}$
b) $\displaystyle\lim_{x\to -3}\frac{x^2+2x-3}{x+3}$

Solution

a) $\displaystyle\lim_{x\to 3}\frac{x^2+2x-3}{x+3}=\frac{3^2+2\times 3-3}{3+3}=2$
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April 9, 2010