A reason you might want 0° to be something other than 0 or 1 As the number x increases through positive values, the numbers 1/x and 1/(In.x) both approach zero. What happens to the number 1/Inx) fx) = (G) as x increases? Here are two ways to find out. a. Evaluate f for x = 10, 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see? b. Graph f in a variety of graphing windows, including win- dows that contain the origin. What do you see? Trace the y-values along the graph. What do you find?

A reason you might want 0° to be something other than 0 or 1 As the number x increases through positive values, the numbers 1/x and 1/(In.x) both approach zero. What happens to the number 1/Inx) fx) = (G) as x increases? Here are two ways to find out. a. Evaluate f for x = 10, 100, 1000, and so on as far as your calculator can reasonably go. What pattern do you see? b. Graph f in a variety of graphing windows, including win- dows that contain the origin. What do you see? Trace the y-values along the graph. What do you find?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section6.3: Trigonometric Functions Of Real Numbers

Problem 67E

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Question

A reason you might want 00 to be something other than 0 or 1 As the number x increases through positive values, the numbers 1>x and 1 > (ln x) both approach zero. What happens to the number

A reason you might want 0° to be something other than 0 or 1
As the number x increases through positive values, the numbers
1/x and 1/(In.x) both approach zero. What happens to the number
1/Inx)
fx) = (G)
as x increases? Here are two ways to find out.
a. Evaluate f for x = 10, 100, 1000, and so on as far as your
calculator can reasonably go. What pattern do you see?
b. Graph f in a variety of graphing windows, including win-
dows that contain the origin. What do you see? Trace the
y-values along the graph. What do you find?

Expert Solution

Step 1

The given function is $f (x) = {(\frac{1}{x})}^{1 / \ln x}$ .

(a) To Find: values of function for x=10,100,1000,... and find the pattern.

(b) To Plot: the graph of this function and write the observation.

Step 2

The given function is $f (x) = {(\frac{1}{x})}^{1 / \ln x}$ .

At $x = 10$ , $f (x) = 0.368$ .

At x=100, $f (x) = 0.368$ .

At x=1000, $f (x) = 0.368$ .

At x=10,000, $f (x) = 0.368$ .

At x=10,0000, $f (x) = 0.368$ .

Here we see that value of the function remain same as we increase x.

So, $f (x) \to 0.368 as x \to \infty$ .

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