Squeeze Theorem Worksheet

Squeeze Theorem Worksheet

Read each question carefully and answer to the best of your ability. Show all your work for full credit.

1. The Squeeze Theorem is also known as the:

2. For the Squeeze Theorem to apply, if g(x) ≤ f(x) ≤ h(x), what must be true about the limits of g(x) and h(x) as x approaches c?

1. The Squeeze Theorem is used to find the limit of a function that is   between two other functions.

2. If \( \lim_{x \to c} g(x) = L \) and \( \lim_{x \to c} h(x) = L \), and if \( g(x) \le f(x) \le h(x) \) for all x in an open interval containing c (except possibly at c itself), then \( \lim_{x \to c} f(x) = \)  .

1. State the Squeeze Theorem in your own words.

Consider the function \( f(x) = x^2 \sin(\frac{1}{x}) \). We want to evaluate \( \lim_{x \to 0} x^2 \sin(\frac{1}{x}) \).

a) What are the upper and lower bounds for \( \sin(\frac{1}{x}) \)?

b) Multiply the inequalities from part (a) by \( x^2 \). What are the new inequalities?

c) Evaluate the limits of the bounding functions as \( x \to 0 \).

d) Use the Squeeze Theorem to find \( \lim_{x \to 0} x^2 \sin(\frac{1}{x}) \).

1. The Squeeze Theorem can only be applied when the limit is approaching zero.

Given that \( 1 - \frac{x^2}{4} \le f(x) \le 1 + \frac{x^2}{2} \) for all x near 0, find \( \lim_{x \to 0} f(x) \).