1. Limits and Continuity

Suppose x lies in the interval (1, 3) with x≠2. Find the smallest positive value of δ such that the inequality 0<|x−2|<δ is true.

Which one of the following intervals is not symmetric about x=5?

a.(1, 9)

b.(4, 6)

c.(3, 8)

d.(4.5, 5.5)

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

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 (8x+5)=13

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→4 x^2−16 / x−4=8 (Hint: Factor and simplify.)

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→7 f(x)=9, where f(x)={3x−12 if x≤7

x+2 if x>7

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→0 x^2=0 (Hint: Use the identity √x2=|x|.)

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→2 (x^2+3x)=10

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 |2x|=6 (Hint: Use the inequality ∥a|−|b∥≤|a−b|, which holds for all constants a and b (see Exercise 74).)

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→a (mx+b)=ma+b, for any constants a, b, and m

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

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→5 1/x^2=1/25

Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

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

lim x→1^+ 1 /1 − x=−∞

Let f(x) =x^2−2x+3.

a. For ε=0.25, find the largest value of δ>0 satisfying the statement

|f(x)−2|<ε whenever 0<|x−1|<δ.