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How do you use the Squeeze Theorem to find #lim (x^2)(cos20(pi*x)) # as x approaches zero?

Answer 1

Cameron Alexander

Refer to explanation

W are aware of that

#-1<=cos(20pix)<=1=>-x^2<=x^2cos(20pix)<=x^2#

Consequently, applying the squeeze theorem as

#lim_(x->0)-x^2=lim_(x->0)x^2=0#

then

#lim_(x->0) x^2cos(20pix)=0#

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Answer 2

Samantha Adkison

To use the Squeeze Theorem to find the limit of (x^2)(cos20(pi*x)) as x approaches zero, we need to find two functions that "squeeze" the given function and have the same limit as x approaches zero.

First, we can observe that -1 ≤ cos(20πx) ≤ 1 for all values of x. Therefore, we can multiply the given function by -1 and 1 to create two functions that "squeeze" it:

-x^2 ≤ (x^2)(cos(20πx)) ≤ x^2

Now, we can take the limit as x approaches zero for all three functions:

lim (x approaches 0) -x^2 = 0 lim (x approaches 0) (x^2)(cos(20πx)) = 0 lim (x approaches 0) x^2 = 0

Since all three functions have the same limit of 0 as x approaches zero, we can conclude that the limit of (x^2)(cos(20πx)) as x approaches zero is also 0.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.