1. Limits and Continuity

a. Analyze ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and${\displaystyle\lim_{x\to-\infty}{f(x)}}$ for each function.

$f\left(x\right)=\frac{1+x-2x^2-x^3}{x^2+1}$

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to \infty }{\lim }x\cos \left(x\right)$

Find the following limits or state that they do not exist. Assume a, b, c, and k are fixed real numbers.

$\underset{x\to 0}{\lim }\frac{1-\cos \left(x\right)}{{\cos }^2\left(x\right)-3\cos \left(x\right)+2}$

Evaluate each limit. 

$\underset{x\to \pi }{\lim }\frac{{\cos }^2\left(x\right)+3\cos \left(x\right)+2}{{\cos }^{}\left(x\right)+1}$

Evaluate each limit. 

$\underset{x\to 3}{\lim }\sqrt{x^2+7}$

Evaluate each limit. 

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(x\right)-1}{\sqrt{\sin \left(x\right)}-1}$

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (2e^x + 3) / (e^x + 1)

Find the horizontal asymptotes of each function using limits at infinity.

f(x) = (3e^5x + 7e^6x) / (9e^5x + 14e^6x)

Determine the following limits.

lim x→∞ (2x − 3) / (4x + 10)

Determine the following limits.

lim x→∞ (x⁴ − 1) / (x⁵ + 2)

Determine the following limits.

lim x→∞ (3 tan^-1 x + 2)

Determine the following limits.

lim w→∞ (ln w²) / (ln w³ + 1)

Determine the following limits.

lim x→−∞ e^x sin x

Determine the following limits.

lim x→∞ (5 + (cos⁴ x) / (x² + x + 1))

A sequence is an infinite, ordered list of numbers that is often defined by a function. For example, the sequence {2,4,6,8,…} is specified by the function f(n) = 2n, where n=1,2,3,….The limit of such a sequence is lim n→∞ f(n), provided the limit exists. All the limit laws for limits at infinity may be applied to limits of sequences. Find the limit of the following sequences or state that the limit does not exist. 

{0,1/2,2/3,3/4,…}, which is defined by f(n) = (n−1) / n, for n=1,2,3,…