Squeeze Theorem Worksheet
A Grade 11 Math worksheet focusing on understanding and applying the Squeeze Theorem to evaluate limits, with various question types.
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Squeeze Theorem Worksheet
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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:
Sandwich Theorem
Pinch Theorem
Both a and b
None of the above
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?
They must be equal
They must be different
They must approach infinity
They must approach zero
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.
True
False
Given that \( 1 - \frac{x^2}{4} \le f(x) \le 1 + \frac{x^2}{2} \) for all x near 0, find \( \lim_{x \to 0} f(x) \).