To prove: The value of .
The Squeeze Theorem
“If when x is near a (except possibly at a) and , then .”
Suppose that c is a constant and the limits and exist, then
Limit law 3:
Limit law 10: where n is a positive integer, if n is even, assume that .
It is trivial that, the value of does not exist.
Thus, the limit of the function does not exist.
Apply the Squeeze Theorem and obtain a function f smaller than and a function h bigger than such that both the functions and approaches 0.
Since the sine function lies between and 1,consider .
Take the exponential of the inequality,
Any inequality remains true when multiplied by a positive number. Since for all x, multiply each side of the inequalities by .
When x approaches to zero, the inequality becomes,
Obtain the value of as follows.
Obtain the value of .
Let , and .
Sketch the graph of the functions , and by using the online graphing calculator as shown below in Figure (1).
From Figure 1, it is observed that and .
If when x approaches 0 and , then by Squeeze Theorem the limit of the function is zero.
That is, .
Hence the required proof is obtained.
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