1. Limits and Continuity

Let $g\(\left\)(x\(\right\))=\(\frac{x^3-4x}{8\left|x-2\right|}\)$. <IMAGE>

Calculate $g\(\left\)(x\(\right\))$ for each value of $x$ in the following table.

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.

limₕ→₀ (1 + 2h)^{1/h}

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. 

$f\left(x\right)=-3e^{-x}$

Use the graph of $f\(\left\)(x\(\right\))$to estimate the value of the limit or state that it does not exist (DNE).

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

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

h(x) = (−5 + (7/x))/(3 – (1/x²))

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

a. How does the graph behave as x → 0⁺?

Give reasons for your answers.

y = (3/2)(x − (1 / x))²/³

Estimating Limits

[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.

Let G(x)=(x + 6)/(x² + 4x − 12)

b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.

Estimate lim θ→0 sin 2θ / sin θ using the graph in part (a).

Use formal definitions to prove the limit statements in Exercises 93–96.

lim x → 0 (−1 / x²) = −∞

Domains and Asymptotes

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

y = (√(x² + 4)) / x

Infinite Limits

Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.

lim x→0 (−1) / (x² (x + 1))

Finding Deltas Algebraically

Each of Exercises 15–30 gives a function f(x) and numbers L, c, and ε>0. In each case, find the largest open interval about c on which the inequality |f(x)−L| <ε holds. Then give a value for δ>0 such that for all x satisfying 0 < |x−c| < δ, the inequality |f(x)−L| < ε holds.

f(x) = 1/x, L = 1/4, c = 4, ε = 0.05

Use the graph of $g\left(x\right)$ in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

$\(\lim\)_{x\(\to\)2}g\(\left\)(x\(\right\))$

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

b. limx→2 f(x)=2

Use an appropriate limit definition to prove the following limits.

lim x→ 5x^2 − 25 / x − 5=10