1. Limits and Continuity

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,…

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. 

{2,3/4,4/9,5/16,…}, which is defined by f(n) = (n+1) / n^2, for n=1,2,3,…

Assume postage for sending a first-class letter in the United States is $0.47 for the first ounce (up to and including 1 oz) plus $0.21 for each additional ounce (up to and including each additional ounce).

b. Evaluate lim w→2.3 f(w).

Determine the following limits.

lim u→0^+ u − 1 / sin u

Determine the following limits.

lim x→0^− 2 / tan x

Complete the following steps for the given functions. 

c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f\left(x\right)=\frac{x^2-2x+5}{3x-2}$

Complete the following steps for the given functions. 

c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$