Mixed limit exercises 1

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

$\lim _{x \to 5} \frac {x^2-25}{x-5} = \answer {10}$

Let $F(z) = \frac {-5 z^3-3}{z^3-5 z^2-2 z}.$ Compute

(a): $\displaystyle \lim _{z\to \infty } F(z) \begin {prompt} = \answer {-5}\end {prompt}$
(b): $\displaystyle \lim _{z\to -\infty }F(z) \begin {prompt} = \answer {-5}\end {prompt}$

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

$\lim _{x \to 2} \frac {\frac {1}{2}-\frac {1}{x}}{2-x} = \answer {-1/4}$

$\lim _{x \to 0} \frac {\sin (2x)}{3x} = \answer {2/3}$

Recall that $\displaystyle \lim _{x \to 0} \frac {\sin x}{x} = 1$ .

$g(x) = \begin {cases} \frac {x^2 - 4}{x-2} &\text {if $x<1$,} \\ -x+1 &\text {if $x>1$.} \end {cases}$ Does $\lim _{x \to 2} g(x)$ exist? If it does, give its value. Otherwise, write DNE.

$\lim _{x \to 2} g(x) = \answer {-1}$

Does $\lim _{x \to 1} g(x)$ exist? If it does, give its value. Otherwise, write DNE.

$\lim _{x \to 1} g(x) = \answer {DNE}$

Consider $f(\psi ) = \frac {-5}{\psi -4}.$ Compute

(a): $\displaystyle \lim _{\psi \to 4^- } f(\psi ) \begin {prompt} = \answer {\infty }\end {prompt}$
(b): $\displaystyle \lim _{\psi \to 4^+ } f(\psi ) \begin {prompt} = \answer {-\infty }\end {prompt}$
(c): $\displaystyle \lim _{\psi \to 4 } f(\psi ) \begin {prompt} = \answer {DNE}\end {prompt}$

Find $\lim _{x\to 6}\left (\frac {\left |6-x\right |}{6-x}\right ) = \answer {DNE}$

Absolute value = piece-wise in disguise!