Mixed limit exercises 2

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

Suppose that $\displaystyle \lim _{u\to 2}c(u)=-5$ , $\displaystyle \lim _{u\to 2}B(u)=-1$ , and $\displaystyle \lim _{u\to 2}Y(u)=-4$ . Compute the limit

$\lim _{u\to 2 } \frac {c(u)}{B(u)-Y(u)}\begin {prompt} = \answer {-\frac {5}{3}}\end {prompt}$

$\lim _{x \to 0} \frac {\tan (x)}{\sin (x)} = \answer {1}$

$\lim _{\psi \to 4 } \frac {\sqrt {\psi +5}-3}{\psi -4}\begin {prompt} = \answer {\frac {1}{6}}\end {prompt}$

$\lim _{x \to 0} \frac {\cos (x)-1}{x} = \answer {0}$

This is a limit that you may have seen before. Alternatively, if you don’t remember seeing this limit before, you can multiply by $\cos (x)+1$ in the numerator and denominator and then apply the Pythagorean identity to get started.

$k(x) = \begin {cases} \frac {\sqrt {x+3}-2}{x-1} & x<1 \\ 7x & x=1 \\ \frac {2x^3 -x}{4x} & x > 1 \end {cases}$ Does $\lim _{x \to 1} k(x)$ exist? If it does, give its value. Otherwise, write DNE.

$\lim _{x \to 1} k(x) = \answer {\frac {1}{4}}$

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

(a): $\displaystyle \lim _{z\to \infty } y(z) \begin {prompt} = \answer {0}\end {prompt}$
(b): $\displaystyle \lim _{z\to -\infty }y(z) \begin {prompt} = \answer {0}\end {prompt}$

Find $\lim _{\theta \to \infty }\left (\cos (\theta )\right ) = \answer {DNE}$

$\lim _{q \to 0} \frac {\frac {1}{7}-\frac {1}{3q}}{\frac {1}{3q}-\frac {1}{7}} = \answer {-1}$