Mixed limit exercises 4

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

$\lim _{z \to -5} \frac {3z}{\frac {1}{z+5}-\frac {2}{z+10}} = \answer {0}$

$z(x) = \frac {3|x|+|x|^3}{2|x|^3 - |x|^2}$

Evaluate the following limits.

$\displaystyle \lim _{x \to \infty } z(x) = \answer {\frac {1}{2}}$
$\displaystyle \lim _{x \to -\infty } z(x) = \answer {\frac {1}{2}}$

$\lim _{x \to -4} \frac {x^2+3x-4}{x^2-4x-32} = \answer {5/12}$

Find $\lim _{\alpha \to 0} \sin \left (\frac {3}{\alpha }\right ) = \answer {DNE}$

Think very carefully how the function $\sin \left (\frac {3}{\alpha }\right )$ behaves as $x$ approaches 0. (You may remember a function similar to this that you explored in lab $\# 1$ .)

$\lim _{x \to \pi /3} \sec ^2(x)+1 = \answer {5}$

Let $r(k) = \frac {\sqrt {k^6+5}-2 k^2}{k-k^3}.$ Compute

(a): $\displaystyle \lim _{k\to \infty } r(k) \begin {prompt} = \answer {-1}\end {prompt}$
(b): $\displaystyle \lim _{k\to -\infty }r(k) \begin {prompt} = \answer {1}\end {prompt}$

Multiply by $\frac {\frac {1}{k^3}}{\frac {1}{k^3}}$

$\lim _{t\to 3 } \frac {t-3}{\sqrt {t-2}-1} = \answer {2}$

Suppose that $\lim _{z\to 5}G(z)=3$ , $\lim _{z\to 5}C(z)=-4$ , and $\lim _{z\to 5}c(z)=1$ . Compute the limit

$\lim _{z\to 5 } \frac {G(z)}{C(z)-c(z)} = \answer {-\frac {3}{5}}$