To find: 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 cubes formula:
Difference of squares 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 at .
“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 t approaches 2 but .
Simplify by using elementary algebra.
Apply the difference of squares formula in the denominator,
Apply the difference of cubes formula in the numerator,
Since the limit of t approaches to 2 but not equal to 2, cancel the common term 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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