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!

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Type...	...to get
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text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
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sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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