1. Limits and Continuity

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (x^4−1)/(x^2−1)

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=√x^2+2x+6−3 / x−1

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=√x^2+2x+6−3 / x−1

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=|1−x^2| / x(x+1)

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=|1−x^2| / x(x+1)

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=3e^x+10 / e^x

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=3e^x+10 / e^x

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

The hyperbolic cosine function, denoted $\cosh \left(x\right)$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh \left(x\right)=\frac{e^x+e^{-x}}{2}$.

a. Determine its end behavior by analyzing $\underset{x\to \infty }{\lim }\cosh \left(x\right)$ and $\underset{x\to -\infty }{\lim }\cosh \left(x\right)$.

Evaluate each limit. 

lim θ→0 (1/(2+sinθ)-1/2)/sin θ

Evaluate each limit. 

lim x→0 cos x−1 / sin^2x

Evaluate each limit. 

lim x→0 e^4x−1 / e^x−1

Evaluate each limit. 

lim x→0+ 1−cos^2x / sin x

Evaluate each limit. 

lim x→e^2 ln^2x−5 ln x+6 lnx−2