Vertical asymptotes occur when

Horizontal asymptotes occur when

Give an example of a function with two vertical asymptotes

Give an example of a function with a horizontal asymptote.

Draw the graph of a function with the following properties: domain = (−5, ∞), range = (−4, ∞) f is continuous everywhere except at 0, 5 f is differentiable everywhere except at 0, 5, 7 f(0) = 4, f′(x) = \frac{2}{3} if x ∈ \{−1\} ∪ [2, 4] f′(1) = 0, f′(x) < 0 if x < −3.

Calculate the following limit. Show all steps.

lim_{x→−∞} \frac{x − 1}{\sqrt{2 − 3x^{2}}}

Express the given quantity as a single log: 3ln(2x) − ln(x + 1) + ln(10x^{2})

Find the exact value of ln(\sqrt{e}) + ln(3e) − ln(3)

FYI: derive quotient rule from product and chain rule:

(\frac{f}{g} )′ = (fg^{−1})′ = f′g^{−1} − fg^{−2}g′ = \frac{f′g − fg′}{g^{2}}