To evaluate: The limit of the function .
The limit of the function is 3.
Suppose that c is a constant and the limits and exist, then
Limit law 8:
Direct substitution property:
If f is a polynomial or a rational function and a is in the domain of f, then .
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 some finite value when both the numerator and the denominator approach to 0.”
By note 2, consider the limit h approaches to zero but .
Simplify by using elementary algebra.
Expand the numerator and simplify further,
Since the limit h approaches zero but not equal to zero, cancel the common term h from both the numerator and the denominator,
By fact 1, if and , then .
Apply the direct substitution property on the limit function.
Thus, the limit of the function is .
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