Question

Consider the function f(x)=(sin(4x))/x. How do you find an interval for x near zero such that the difference between your conjectured limit and the value of the function is less than 0.01?

`f(x)=(sin(4x))/x`

(I can figure out A and B, but I am having trouble figuring out C)

(A)
Fill in the following table of values for f(x):

x= -0.1 -0.01 -0.001 -0.0001 0.0001 0.001 0.01 0.1

(B) Based on your table of values, what would you expect the limit of f(x) as x approaches zero to be?

(C) Graph the function to see if it is consistent with your answers to parts (a) and (b). By graphing, find an interval for x near zero such that the difference between your conjectured limit and the value of the function is less than 0.01. In other words, find a window of height 0.02 such that the graph exits the sides of the window and not the top or bottom. What is the window?

? ≤ x ≤ ?
? ≤ y ≤ ?

Answer 1

A.
B. 4
C. -.10`<=`x `<=` .10
3.99`<=`y`<=`4.01

Explanation:

A. Observe the tables calculated via https://www.desmos.com/calculator.

B. We should expect `4` because let us remember that `sin(theta)/(theta) = 1` We first prove this with the squeeze theorem. not listed

From here note that we can multiple our function by `4/4` resulting in `4(sin(4x)/(4x))` substituting `4x = theta`

`4(sin(theta)/(theta)) = 4(1) = 4`
C.
the absolute value for `abs(-.1-.1) = .2` for -.10`<=` x `<=` .10
the absolute value for `abs(3.99-4.01) = .2`