Mixed limit exercises 3

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Here is an opportunity for you to practice evaluating many different types of limits.

$m(x) = \begin {cases} \frac {1}{x} & x < 0 \\ 1 & x \geq 0 \end {cases}$

Evaluate the following limits if they exist. Otherwise, write DNE.

$\displaystyle \lim _{x \to 0^-} m(x) = \answer {-\infty }$
$\displaystyle \lim _{x \to 0^+} m(x) = \answer {1}$
$\displaystyle \lim _{x \to 0} m(x) = \answer {DNE}$

Find $\displaystyle \lim _{x \to 0} \sqrt [3]{x} \cos \left (\frac {1}{x}\right ) = \answer {0}$

Find $\lim _{x\to 0}\left (\frac {\cos ^{2}(x)-1}{\cos (x)-1}\right ) = \answer {2}$

$\lim _{x \to 0} \frac {\sin (5x)}{\sin (2x)} = \answer {5/2}$

Find $\displaystyle \lim _{x \to -3} \frac {(x+3)^2}{|x+3|} = \answer {0}$

$\lim _{h \to 0} \frac {\sqrt {2+h} - \sqrt {2}}{h} = \answer {1/(2\sqrt {2})}$

Consider $g(a) = \frac {6}{a^2}.$ Compute $\displaystyle \lim _{a \to 0 } g(a) \begin {prompt} = \answer {\infty }\end {prompt}$

$\lim _{\theta \to -1 } \frac {\theta +1}{-2 \theta ^2-6 \theta -4} = \answer {-\frac {1}{2}}$