To evaluate: The limit of the function .
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 some finite value when both the numerator and the denominator approach 0.”
By note 2, consider the limit x approaches to 3 but .
Simplify by using elementary algebra.
Multiply both the numerator and the denominator by the conjugate of the numerator,
Apply the difference of squared formula,
Factorize the numerator,
Substitute for in equation (2),
Since the limit of x approaches 3 but not equal to 3, 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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