Estimate a limit from a table of values on both sides, then confirm it exactly. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
11 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - x - 2}{x^{2} + 4 x - 12} \) at \( \displaystyle x = 2 \pm 0.1,\ 2 \pm 0.01,\ 2 \pm 0.001,\ 2 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 2} f(x) \).
f(2) itself is undefined (0/0), so the table approaches 2 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{\sqrt{x} - 1}{x - 1} \) at \( \displaystyle x = 1 \pm 0.1,\ 1 \pm 0.01,\ 1 \pm 0.001,\ 1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 1} f(x) \).
f(1) itself is undefined (0/0), so the table approaches 1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + 5 x + 4}{x^{2} + 8 x + 7} \) at \( \displaystyle x = -1 \pm 0.1,\ -1 \pm 0.01,\ -1 \pm 0.001,\ -1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to -1} f(x) \).
f(-1) itself is undefined (0/0), so the table approaches -1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + x - 6}{x^{2} + 2 x - 8} \) at \( \displaystyle x = 2 \pm 0.1,\ 2 \pm 0.01,\ 2 \pm 0.001,\ 2 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 2} f(x) \).
f(2) itself is undefined (0/0), so the table approaches 2 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 3 x + 2}{x^{2} + 6 x - 7} \) at \( \displaystyle x = 1 \pm 0.1,\ 1 \pm 0.01,\ 1 \pm 0.001,\ 1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 1} f(x) \).
f(1) itself is undefined (0/0), so the table approaches 1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 2 x - 3}{x^{2} + 4 x + 3} \) at \( \displaystyle x = -1 \pm 0.1,\ -1 \pm 0.01,\ -1 \pm 0.001,\ -1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to -1} f(x) \).
f(-1) itself is undefined (0/0), so the table approaches -1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 1}{x^{2} + 5 x + 4} \) at \( \displaystyle x = -1 \pm 0.1,\ -1 \pm 0.01,\ -1 \pm 0.001,\ -1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to -1} f(x) \).
f(-1) itself is undefined (0/0), so the table approaches -1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 9}{x^{2} + 4 x - 21} \) at \( \displaystyle x = 3 \pm 0.1,\ 3 \pm 0.01,\ 3 \pm 0.001,\ 3 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 3} f(x) \).
f(3) itself is undefined (0/0), so the table approaches 3 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - 2 x - 3}{x^{2} + 5 x - 24} \) at \( \displaystyle x = 3 \pm 0.1,\ 3 \pm 0.01,\ 3 \pm 0.001,\ 3 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 3} f(x) \).
f(3) itself is undefined (0/0), so the table approaches 3 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} + 2 x - 3}{x^{2} + 5 x - 6} \) at \( \displaystyle x = 1 \pm 0.1,\ 1 \pm 0.01,\ 1 \pm 0.001,\ 1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to 1} f(x) \).
f(1) itself is undefined (0/0), so the table approaches 1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Make a table of values of \( \displaystyle f(x) = \frac{x^{2} - x - 2}{x^{2} + 5 x + 4} \) at \( \displaystyle x = -1 \pm 0.1,\ -1 \pm 0.01,\ -1 \pm 0.001,\ -1 \pm 0.0001 \) and use it to estimate \( \displaystyle \lim_{x \to -1} f(x) \).
f(-1) itself is undefined (0/0), so the table approaches -1 from both sides.
✓ Nihil obstat Lines: 1 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify