1. Limits and Continuity

Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)

19. lim(x→∞)arccsc(x)

Determine the following limits.

lim x→1000 18π^2

Evaluate each limit and justify your answer. 

lim x→1 (x+5x / x+2)^4

Limits and Continuity

Suppose that ƒ(t) and ƒ(t) are defined for all t and that lim t → t₀ ƒ(t) = ―7 and lim (t → t₀) g (t) = 0 . Find the limit as t → t₀ of the following functions.

e. cos (g(t))

Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)

Let f(x) = {x^2+1 / if x<−1

√x+1 if x≥−1.

Compute the following limits or state that they do not exist.

limx→−1 f(x)

Find the limit.

$\(\lim\)_{x\(\rarr\)0}x^3+5x^2-7x+3$

Use Theorem 3.10 to evaluate the following limits.

lim x🠂0 sin ax / sin bx, where a and b are constants with b ≠ 0.

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Determine the following limits.

b. $\underset{x\to 2}{\lim }\frac{x-2}{x^5-4x^3}$

Limits with trigonometric functions

Find the limits in Exercises 43–50.

lim x→0 tan x

Find all vertical asymptotes $x=a$ of the following functions. For each value of $a$, determine $\underset{x\to a^{+}}{\lim }f\left(x\right)$, $\underset{x\to a^{-}}{\lim }f\left(x\right)$, and $\underset{x\to a}{\lim }f\left(x\right)$.

$f\left(x\right)=\frac{\cos \left(x\right)}{x^2+2x}$

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→3 1/ x − 3(1 /√x + 1 − 1/2)

Determine the following limits. 

lim x→∞ x^4+7 / x^5+x^2−x