1. Limits and Continuity

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

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g. limx→0+ f(x) = limx→0− f(x)

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. 

$f\left(x\right)=\sin x$

For any real number x, the floor function (or greatest integer function) ⌊x⌋ is the greatest integer less than or equal to x (see figure).

a. Compute lim x→−1^− ⌊x⌋, lim x→−1^+ ⌊x⌋,lim x→2^− ⌊x⌋, and lim x→2^+ ⌊x⌋.

Finding Limits Graphically

Which of the following statements about the function y = f(x) graphed here are true, and which are false?

a. limx→−1+ f(x) = 1

Limits as x → ∞ or x → −∞

The process by which we determine limits of rational functions applies equally well to ratios containing noninteger or negative powers of x. Divide numerator and denominator by the highest power of x in the denominator and proceed from there. Find the limits in Exercises 23–36. Write ∞ or −∞ where appropriate.

lim x→∞ (x − 3) / √(4x² + 25)

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (x + 1)/(x² + 3)

Finding Limits of Differences When x → ±∞

Find the limits in Exercises 84–90. (Hint: Try multiplying and dividing by the conjugate.)

lim x → ∞ (√(x + 9) − √(x + 4))

Given the graph of the function $g\left(x\right)$ below, on which interval is $g\left(x\right)$ continuous?

If a function f represents a system that varies in time, the existence of lim ${\(\displaystyle\[\lim\)_{t\(\rightarrow\]\infty\)}{f(t)}}$ means that the system reaches a steady state (or equilibrium). For the following systems, determine whether a steady state exists and give the steady-state value.

The population of a colony of squirrels is given by $p\left(t\right)=\frac{1500}{3+2e^{-0.1t}}$.

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx→1 f(x) = 1 if f(x) = {x², x ≠ 1

2, x = 1

Analyze the following limits. Then sketch a graph of y=tanx with the window [−π,π]×[−10,10]and use your graph to check your work.

lim x→π/2^− tan x

Which of the following regions in the plane has an area equal to the limit $\underset{n\to \infty }{lim}\frac{1}{n}(1+\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\dots +\frac{1}{n^{n-1}})$? Do not evaluate the limit.

Given the geometric series $\sum n_0=\infty x^n$, find the sum of the series $\sum n_1=\infty n{x}^{n-1}$, where $\left|x\right|<1$.

Use the ratio test to determine whether the series $\sum _{n=1}^{\infty }\frac{\left(n\right)!}{n^n}$ converges or diverges.