1. Limits and Continuity

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim t→2+ |2t − 4|t^2 − 4

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→3 x − 3 /|x − 3|

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

If limx→a f(x) = L, then f(a)=L.

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers.

The limit lim x→a f(x) / g(x) does not exist if g(a)=0.

Evaluate lim x→2^+ √x−2.

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