1. Limits and Continuity

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→3 x^3=27

Determine the following limits at infinity.

lim x→∞ (3+10/x^2)

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Find the limit by creating a table of values.

$\(\lim\)_{x\(\rarr\)0}-4x+2$

Complete the following steps for the given functions. 

b. Find the vertical asymptotes of f (if any).

$f\left(x\right)=\frac{3x^2-2x+5}{3x+4}$

Use the graph of $h$ in the figure to find the following values or state that they do not exist. <IMAGE>

${\(\displaystyle\]\lim\)_{x\(\to\)4}h\(\left\)(x\(\right\))}$

Use the graph of $f$ in the figure to find the following values or state that they do not exist. <IMAGE>

$f\(\left\)(1\(\right\))$

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\(\rarr\)0}f\(\left\)(x\(\right\))$

The following table gives the position $s\(\left\)(t\(\right\))$ of an object moving along a line at time $t$. Determine the average velocities over the time intervals $\(\left\[\lbrack\)1,1.01\(\right\]\rbrack\)$, $\(\left\[\lbrack\)1,1.001\(\right\]\rbrack\)$, and $\(\left\]\lbrack\)1,1.0001^{}\(\right\).]$. Then make a conjecture about the value of the instantaneous velocity at $t=1$. <IMAGE>

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

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

Let $g\(\left\)(t\(\right\))=\(\frac{t-9}{\sqrt{t}\)-3}$.

Make two tables, one showing values of $g$ for $t=8.9,8.99$, and $8.999$ and one showing values of $g$ for $t=9.1,9.01$, and $9.001$.

Using the Formal Definition

Each of Exercises 31–36 gives a function f(x), a point c, and a positive number ε. Find L = lim x→c f(x). Then find a number δ > 0 such that |f(x)−L| < ε whenever 0 < |x−c| < δ.

f(x) = −3x − 2, c = −1, ε = 0.03

Finding Limits of Differences When x → ±∞

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

lim x → −∞ (2x + √(4x² + 3x − 2))

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−x², P(2,3)

Using the Sandwich Theorem

If 2−x² ≤ g(x) ≤ 2cosx for all x, find limx→0 g(x).