Explanation Given information: The expression, lim x → 0 x 2 cos ( 1 x 2 ) . Formula used: Squeeze Theorem states, consider three functions l ( x ) , m ( x ) and n ( x ) which satisfy the relation l ( x ) ≤ m ( x ) ≤ n ( x ) . If lim x → c l ( x ) = A and lim x → c n ( x ) = A then lim x → c m ( x ) = A , that is m ( x ) is squeezed between l ( x ) and n ( x ) and its limit is same as the two functions near that point. Proof: Consider the expression, lim x → 0 x 2 cos ( 1 x 2 ) . Recall that range of cosine function lies between the interval [ − 1 , 1 ] that is mathematically, − 1 ≤ cos x ≤ 1 Or, − 1 ≤ cos ( 1 x 2 ) ≤ 1 Multiply the above equation by x 2 , − x 2 ≤ x 2 cos ( 1 x 2 ) ≤ x 2 Compare the above equation with Squeeze Theorem which states, consider three functions l ( x ) , m ( x ) and n ( x ) which satisfy the relation l ( x ) ≤ m ( x ) ≤ n ( x )

15th Edition

ISBN: 9781305420878

Author: Stewart

Publisher: CENGAGE L

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 2, Problem 30E

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To prove: The value of the expression limx→0x2cos(1x2)=0 .

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Chapter 2, Problem 29E

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Chapter 2 Solutions

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To prove: The value of the expression lim x → 0 x 2 cos ( 1 x 2 ) = 0 .

Solution Summary: The author explains the formula used to prove the value of the expression undersetxto 0mathrm

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To prove: The value of the expression limx→0x2cos(1x2)=0 .

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Chapter 2 Solutions