To estimate: The value of the function to two decimal places when x approaches zero by using the graph of the function .
The estimated value of the function when x approaches zero approaches to 0.29.
The function .
Use the online graphing calculator to draw the graph of the function .
From the graph, as , then is not defined. But x approaches 0, and then tends to 0.289.
That is, .
Thus, the estimated value of is .
To estimate: The value of the limit to four decimal places when x close to 0 by using the table of values of .
The estimated value of the limit by using the table of values of for x close to 0 is .
Construct the table of values of for x close to 0.
From the table, approaches 0.2887 as x gets more close to 0.
That is, .
Thus, the limit appears to be approximately equal to .
To find: The exact limit value of the function as x approaches 0.
The exact limit value of the function is .
The limit of the function as x approaches 0 is .
Suppose that c is a constant and the limits and exist, then
Limit law 1:
Limit law 2:
Limit law 3:
Limit law 4:
Limit law 5: if
Limit law 7:
Limit law 8:
Limit law 11: where n is a positive integer, if n is even, assume that .
Direct substitution property:
If f is a polynomial or a rational function and a is in the domain of f, then .
Difference of square formula:
If when , then , provided the limit exist.
The direct substitution method is not applicable for the function as the function is in indeterminate form when .
The Quotient rule is not applicable for the function as the limit of the denominator is zero.
“The limit may be infinite or some finite value when both the numerator and the denominator approach 0.”
By note 3, take the limit x approaches 0 but .
Simplify by using elementary algebra, .
Take the conjugate of the numerator and multiply and divide by .
Apply the formula for the difference of square,
Since the limit x approaches 0 but not equal to 0, cancel the common term from both the numerator and the denominator,
Use fact 1, and , then .
Use the limit laws to find the required limit function.
Thus, the exact limit of the function is .
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