1. Limits and Continuity

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}$

Suppose $\underset{x\to a}{\lim }f\left(x\right)=L$ and $\underset{x\to a}{\lim }g\left(x\right)=M$. Prove that $\underset{x\to a}{\lim }(f\left(x\right)-g\left(x\right))=L-M$.

Finding Limits Graphically

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

e. limx→0 f(x) exists

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

Make a conjecture about the value of ${\(\displaystyle\]\lim\)_{x\(\to\)2}\(\frac{x^2-4}{x-2}\)}$.

Finding Deltas Graphically

In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.

Slope of a Curve at a Point

In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.

y=√7−x, P(−2,3)

Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 sin θ / sin 2θ

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→∞ √(x² + 1) / (x + 1)

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

b. From the graph of the position function, identify the time at which the projectile has an instantaneous velocity of zero; call this time t=a.

A projectile is fired vertically upward and has a position given by s(t)=−16t^2+128t+192, for 0≤t≤9.

d. For what values of t on the interval [0, 9] is the instantaneous velocity positive (the projectile moves upward)?

Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why. <IMAGE>

${\lim }_{x\to 1^{-}}f\left(x\right)$

The table gives the position s(t)of an object moving along a line at time t, over a two-second interval. Find the average velocity of the object over the following intervals. <IMAGE>

a. $[0, 2]$

Finding Limits Graphically

Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2

<IMAGE>

c. Find limx→−1− f(x) and limx→−1+ f(x).

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

b. Extending the table if necessary, try to determine the rate of change of F(x) at 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) = mx, m > 0, L = 2m, c = 2, ε = 0.03