1. Limits and Continuity

The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>

lim x→1 f(x)

Find the limit: $\underset{t\to 5}{lim}\frac{5+t^2}{5-t^2}$

Find the limits in Exercises 53–58. Write ∞ or −∞ where appropriate.

lim (x² − 3x + 2) / (x³ − 2x²) as

a. x→0⁺

{Use of Tech} Population growth Consider the following population functions.

d. Evaluate and interpret lim t→∞ p(t).

p(t) = 600 (t²+3/t²+9)

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

Determine the following limits.

lim x→5 x − 7 / x(x − 5)^2

9. True, or false? As x→∞,

a. x = o(x)

Determine the following limits.

c. lim x→−2 (x − 4) / x(x + 2)

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,…

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

lim x→4 3(x − 4)√x + 5 / 3 − √x + 5

52-56. In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations.

where P(t) is the population, for t ≥ 0, and r > 0 and K > 0 are given constants.

d. Find lim(t→∞) P(t) and check that the result is consistent with the graph in part (c).

Calculating Limits

Find the limits in Exercises 11–22.

limt→6 8(t−5)(t−7)

Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

Determine the following limits.

a. lim x→2^+ 1 x − 2

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.

f. | ƒ(t) |