To find: The value of .
The limit of the function is .
Direct substitution property:
If f is a polynomial or a rational function and a is in the domain of f, then .
Difference of squared formula:
If when , then , provided the limit exist.
The direct substitution method is not applicable for the function as the function is in an indeterminate form when .
“The limit may be infinite or it may be some finite value when both the numerator and the denominator approach 0.”
By note 2, consider the limit u approaches 1 but .
Simplify by using elementary algebra as follows.
Apply the difference of squared formula,
Again, apply the difference of squared formula in the numerator,
Factorize the denominator of ,
Substitute for in equation (2),
Since the limit u approaches 1 but not equal to 1, cancel the common term from both the numerator and the denominator,
Use fact 1, and , then .
Apply the direct substitution property on the limit function.
Thus, the limit of the function is .
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