To estimate: The value of the function when x approaches zero by graphing the function .
The estimated value of the function when x approaches zero approaches 0.6667 or .
The graph of the function .
Draw the graph of the function by using the graphing calculator as shown below in Figure 1.
From the graph, as , then is not defined. But x approaches 0, then goes to .
That is, .
Thus, the estimated value of is .
To guess: The value of the limit by using the table of values of for x close to 0.
The value of the limit by using the table of values of for x close to 0 is .
Make the table of values of for x close to 0.
From the table, as x gets more close to 0, the value of approaches 0.66667 or .
That is, .
Thus, the limit appears to be .
To prove: The 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 6: where n is a positive integer
Limit law 7:
Limit law 8:
Limit law 9: where n is a positive integer.
Limit law 10: where n is a positive integer, if n is even, assume that .
Limit law 11: where n is a positive integer, if n is even, assume that .
The Quotient rule is not applicable directly for the function because the limit of the denominator is zero.
“The limit may be infinite or it may be some finite value when both numerator and denominator approach 0.”
By note 3, take the limit x approaches 0 but .
Simplify by using elementary algebra, .
Take the conjugate of the denominator and multiply and divide of .
Use the difference of square formula,
Since the limit x approaches 0 but not equal to 0, cancel the common term of both numerator and denominator,
Use fact 1, and , then
Use the limit laws to obtain the limit of the function as below:
Thus, the limit of the function is .
Hence the required proof is obtained.
Subscribe to bartleby learn! Ask subject matter experts 30 homework questions each month. Plus, you’ll have access to millions of step-by-step textbook answers!