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AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 1

Lesson 9: Determining limits using the squeeze theorem

Squeeze theorem

AP.CALC: LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.2 (EK)

We want to find limit, start subscript, x, \to, 0, end subscript, start fraction, x, divided by, start text, s, i, n, end text, left parenthesis, x, right parenthesis, end fraction. Direct substitution and other algebraic methods don't seem to work.

Looking at the graph of start color #11accd, f, left parenthesis, x, right parenthesis, equals, start fraction, x, divided by, start text, s, i, n, end text, left parenthesis, x, right parenthesis, end fraction, end color #11accd, we can estimate that the limit is equal to 1.

To prove that limit, start subscript, x, \to, 0, end subscript, start fraction, x, divided by, start text, s, i, n, end text, left parenthesis, x, right parenthesis, end fraction, equals, 1, we can use the squeeze theorem.

Luke suggested that we use the functions start color #e07d10, g, left parenthesis, x, right parenthesis, equals, x, plus, 1, end color #e07d10 and start color #ca337c, h, left parenthesis, x, right parenthesis, equals, minus, x, plus, 1, end color #ca337c in order to apply the squeeze theorem.

Does Luke's suggestion seem to be correct?

(Choice A)
Yes, Luke's suggestion seems to be correct.
(Choice B)
No, Luke's suggestion is incorrect because it's not true that the output of one function is always below f, left parenthesis, x, right parenthesis and the output of the other function is always above it for x-values near 0.
(Choice C)
No, Luke's suggestion is incorrect because it's not true that the limits of start color #e07d10, g, end color #e07d10 and start color #ca337c, h, end color #ca337c are both equal to 1.

Stuck?

AP®︎/College Calculus AB

Course: AP®︎/College Calculus AB > Unit 1

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