1. Limits and Continuity

[Technology Exercise] Graph the functions in Exercises 113 and 114. Then answer the following questions.

b. How does the graph behave as x → ±∞?

Give reasons for your answers.

y = (3/2)(x / (x − 1))²/³

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

limx →4 (9 − x) = 5

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. 

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

Using the Formal Definition

Prove the limit statements in Exercises 37–50.

lim x→0 x sin (1/x) = 0

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Domains and Asymptotes

Determine the domain of each function in Exercises 69–72. Then use various limits to find the asymptotes.

y = 4 + 3x² / (x² + 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

Suppose f(x) lies in the interval (2, 6). What is the smallest value of ε such that |f (x)−4|<ε?

Use the precise definition of a limit to prove the following limits. Specify a relationship between ε and δ that guarantees the limit exists.

lim x→1 x^4=1

Consider the graph of y=cot^−1 x(see Section 1.4) and determine the following limits using the graph.

lim x→∞ cot^−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→⁻∞ (³√x − ⁵√x) / (³√x + ⁵√x)

Find the limit by creating a table of values.

$\lim_{x\rarr0}-4x+2$

Find the limit by creating a table of values.

$\lim_{x\rarr2}3x^2+5x+1$

Find the limit by creating a table of values.

$\lim_{x\rarr1}\frac{x^2-4}{x-2}$

Find the limit using the graph of $f\left(x\right)$shown.

$\lim_{x\rarr1}f\left(x\right)$

Find the limit using the graph of $f\left(x\right)$shown.

$\lim_{x\rarr-2}f\left(x\right)$