   Chapter 2.6, Problem 39E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Find the limit or show that it does not exist. lim x → ∞ ( e − 2 x cos x )

To determine

To find: The value of limx(e2xcosx).

Explanation

Theorem used:

The Squeeze Theorem

“If g(x)f(x)h(x) when x is near a (except possibly at a) and limxag(x)=limxah(x)=L then limxaf(x)=L.”

Calculation:

Obtain the value of the function as x approaches infinity as follows.

Consider the function f(x)=e2xcosx.

Since the range of cosx lies between −1 and 1, consider 1cosx1.

Multiply the above inequality by e2x,

1×e2xe2xcosx1×e2xe2xe2xcosxe2x

So e2xcosx is bounded above by e2x and bounded below by e2x.

Let g(x)=e2x and h(x)=e2x.

Take the limit of upper and lower inequalities as x approaches infinity.

As x goes to infinity, g(x) goes to zero

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