Estimating A Limit

 
Problem Statement

2: {2,4,6,8} 4: eval pow(2, 2) 4.002: eval 4+.002 4.001: eval 4+.001 4.0005: eval 4+.0005 4.0001: eval 4+.0001 0.2499688: eval round((sqrt(4.002)-2)/(4.002-4),7) 0.2499844: eval round((sqrt(4.001)-2)/(4.001-4),7) 0.2499922: eval round((sqrt(4.0005)-2)/(4.0005-4),7) 0.2499984: eval round((sqrt(4.0001)-2)/(4.0001-4),7) 3.998: eval 4-.002 3.999: eval 4-.001 3.9995: eval 4-.0005 3.9999: eval 4-.0001 0.2500313: eval round((sqrt(3.998)-2)/(3.998-4),7) 0.2500156: eval round((sqrt(3.999)-2)/(3.999-4),7) 0.2500078: eval round((sqrt(3.9995)-2)/(3.9995-4),7) 0.2500016: eval round((sqrt(3.9999)-2)/(3.9999-4),7) 0.2500: eval round(1/(2*2),4)

Estimate the limit numerically or state that the limit doesn't exist.

 
Step 1

To determine if a limit exists numerically for , make a table of values of f(x) for x close to c but greater than c (that is, x → c+) and a second table of values of f(x) for x close to c but less than c (that is, x → c–). If both tables indicate convergence to the same number L, we take L to be an estimate for the limit.

In the given problem, c is 4 and f(x) is .

Question Sequence

Question 1

Complete the table of values of f(x) as x → 4+. (Round your answers to seven decimal places.)

x f(x)
4.002 0.2499688
4.001
4.0005
4.0001
_max_tries:2 _feedback_incorrect_first: No. Calculate f(x) for each of the values of x given in the left column, and enter those values in the right column. Remember to round to 7 decimal places. _feedback_incorrect: Incorrect. See above for the correct answers, and try to work out where you went wrong before going on to the next question. _feedback_correct: Nice job. _question_report_text: Values of f(x) for x = c + 0.001 / c + 0.0005 / c + 0.0001

Question 2

As x → 4+, f(x) approaches (rounded to four decimal places)

_max_tries:2 _feedback_incorrect_first: No. What number does f(x) appear to approach as x approaches 4 from above? Remember to round to 4 decimal places. _feedback_incorrect: Incorrect. Do you see where you went wrong? _feedback_correct: Exactly. _question_report_text: As x → c+, f(x) approaches:

 
Step 2

Question Sequence

Question 3

Complete the table of values of f(x) as x → 4–. (Round your answers to seven decimal places.)

x f(x)
3.998 0.2500313
3.999
3.9995
3.9999
_max_tries:2 _feedback_incorrect_first: No. Just like in the previous step, calculate f(x) for each of the values of x given in the left column, and enter those values in the right column. _feedback_incorrect: Incorrect. See above for the correct answers, and try to work out where you went wrong before going on to the next question. _feedback_correct: Perfect. _question_report_text: Values of f(x) for x = c – 0.001 / c – 0.0005 / c – 0.0001

Question 4

As x → 4–, f(x) approaches (rounded to four decimal places)

_max_tries:2 _feedback_incorrect_first: No. What number does f(x) appear to approach as x approaches 4 from below? _feedback_incorrect: Incorrect. Do you see where you went wrong? _feedback_correct: That's it. _question_report_text: As x → c–, f(x) approaches:

 
Step 3

Question 5

Since both tables indicate convergence to the same number 0.2500 as x approaches 4 from the left and from the right, then 0.2500 is an estimate for the limit and we write

= .

_max_tries:2 _feedback_incorrect_first: No. What number did you find that f(x) approached as x approached 4 from above and below? _feedback_incorrect: Incorrect. Do you see where you went wrong? _feedback_correct: Excellent work. _question_report_text: Final estimate of the limit of f(x)