Chapter 1. Estimating A Limit

1.1 Problem Statement

$a: {2,4,6,8} $a2: eval pow($a, 2) $p1_1: eval $a2+.002 $p1_2: eval $a2+.001 $p1_3: eval $a2+.0005 $p1_4: eval $a2+.0001 $p1_1_a: eval round((sqrt($p1_1)-$a)/($p1_1-$a2),7) $p1_2_a: eval round((sqrt($p1_2)-$a)/($p1_2-$a2),7) $p1_3_a: eval round((sqrt($p1_3)-$a)/($p1_3-$a2),7) $p1_4_a: eval round((sqrt($p1_4)-$a)/($p1_4-$a2),7) $p2_1: eval $a2-.002 $p2_2: eval $a2-.001 $p2_3: eval $a2-.0005 $p2_4: eval $a2-.0001 $p2_1_a: eval round((sqrt($p2_1)-$a)/($p2_1-$a2),7) $p2_2_a: eval round((sqrt($p2_2)-$a)/($p2_2-$a2),7) $p2_3_a: eval round((sqrt($p2_3)-$a)/($p2_3-$a2),7) $p2_4_a: eval round((sqrt($p2_4)-$a)/($p2_4-$a2),7) $ans: eval round(1/(2*$a),4)

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

1.2 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 $a2 and f(x) is .

Question Sequence

Question 1.1

Complete the table of values of f(x) as x → $a2+. (Round your answers to seven decimal places.)

x f(x)
$p1_1 $p1_1_a
$p1_2 4KozdQDseRa1gpJK
$p1_3 jY3eeGkeo+oqGJ6H
$p1_4 Bd1SHbUm9tYOoTh7
_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 1.2

As x → $a2+, f(x) approaches qjqZz1N+poo= (rounded to four decimal places)

_max_tries:2 _feedback_incorrect_first: No. What number does f(x) appear to approach as x approaches $a2 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:

1.3 Step 2

Question Sequence

Question 1.3

Complete the table of values of f(x) as x → $a2–. (Round your answers to seven decimal places.)

x f(x)
$p2_1 $p2_1_a
$p2_2 OcwxAtKKG+rgKfFr
$p2_3 O+84XqQLDHys0RUf
$p2_4 DA1vLZke2x8KoTCA
_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 1.4

As x → $a2–, f(x) approaches qjqZz1N+poo= (rounded to four decimal places)

_max_tries:2 _feedback_incorrect_first: No. What number does f(x) appear to approach as x approaches $a2 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:

1.4 Step 3

Question 1.5

Since both tables indicate convergence to the same number $ans as x approaches $a2 from the left and from the right, then $ans is an estimate for the limit and we write

= qjqZz1N+poo= .

_max_tries:2 _feedback_incorrect_first: No. What number did you find that f(x) approached as x approached $a2 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)