1. Limits and Continuity

Explain why lim x→3^+ √ x−3 / 2−x does not exist.

A sine limit It can be shown that 1−x^2/ 6 ≤ sin x/ x ≤1, for x near 0.

Use these inequalities to evaluate lim x→0 sin x/ x.

Suppose g(x) = {x^2−5x if x≤−1

ax^3−7 if x>−1.

Determine a value of the constant a for which lim x→−1 g(x) exists and state the value of the limit, if possible.

Calculate the following limits using the factorization formula x^n−a^n=(x−a)(x^n−1+ax^n−2+a^2x^n−3+⋯+a^n−2x+a^n−1), where n is a positive integer and a is a real number.

lim x→1 x^6 − 1 / x − 1

Sketch a graph of y=2^x and carefully draw three secant lines connecting the points P(0, 1) and Q(x,2^x), for x=−3,−2, and −1.

Evaluate lim x→1 3√x − 1 / x (Hint: x−1=(3√x)^3−1^3.)

Find functions f and g such that lim x→1 f(x)=0 and lim x→1 (f(x)g(x))=5.

Find constants b and c in the polynomial p(x)=x^2+bx+c such that lim x→2 p(x) / x−2=6. Are the constants unique?

Suppose g(x)=f(1−x) for all x, lim x→1^+ f(x)=4, and lim x→1^− f(x)=6. Find lim x→0^+ g(x) and lim x→0^− g(x).

Evaluate each limit. 

lim x→2 √4x+10 / 2x−2

Evaluate each limit. 

lim x→−1 (x^2−4+ 3√x^2−9)

Evaluate lim x→1 (x^3+3x^2−3x+1).

Suppose the rental cost for a snowboard is $25 for the first day (or any part of the first day) plus $15 for each additional day (or any part of a day).

Evaluate lim t→2.9 f(t).

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 (4f(x))

Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.

lim x→1 (f(x)−g(x))