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Amy Ferguson Moncure

Calculus BC ยท Unit 1

Estimate a Limit from a Table

Use both sides of a table. A function value, an average of the two sides, or one huge output is not automatically the limit.

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Key Ideas Easy Medium Hard

\(\text{left-hand values and right-hand values} \to L\)

then the estimate of the two-sided limit is L

What to Remember

A table is evidence, not a proof

If the y-values get close to one number from both sides, that number is the best estimate of the limit. A single output in the table is not the limit.

Check the two sides separately

If the left-hand values and the right-hand values approach different numbers, the two-sided limit does not exist. Do not average those two numbers.

The function value is a separate box

If the table lists \(f(a)\), compare it with the limit. Continuity needs those to be equal. If \(f(a)\) is missing, the table cannot show continuity.

Unbounded growth

If the outputs grow without a ceiling as x approaches a from one side, estimate that one-sided limit as \(\infty\) or \(-\infty\).

Level 1

Easy

Say what the numbers are approaching.

Example
The table gives values of f, and \(f(2) = 0\). Estimate \(\lim_{x \to 2} f(x).\)
\(x\)1.91.991.9992.0012.012.1
\(f(x)\)7.27.027.0026.9986.986.8

From the left, the values 7.2, 7.02, 7.002 approach 7.

From the right, the values 6.998, 6.98, 6.8 approach 7.

Both sides approach the same number.

\(\lim_{x \to 2} f(x) = 7\)

\(f(2) = 0\) is not the limit.

Answer: \(7\)

1
Use the same table, with \(f(2) = 0\). Find \(f(2).\)
\(x\)1.91.991.9992.0012.012.1
\(f(x)\)7.27.027.0026.9986.986.8
Show Answer

The prompt gives the function value.

\(f(2) = 0\)

Answer: \(0\)

2
Estimate \(\lim_{x \to 3^-} f(x).\)
\(x\)2.92.992.9993.0013.013.1
\(f(x)\)5.15.015.0011.0021.021.2
Show Answer

Use only the x-values less than 3: 5.1, 5.01, 5.001.

Those values approach 5.

\(\lim_{x \to 3^-} f(x) = 5\)

Answer: \(5\)

3
Estimate \(\lim_{x \to 3^+} f(x).\)
\(x\)2.92.992.9993.0013.013.1
\(f(x)\)5.15.015.0011.0021.021.2
Show Answer

Use only the x-values greater than 3: 1.002, 1.02, 1.2.

Those values approach 1.

\(\lim_{x \to 3^+} f(x) = 1\)

Answer: \(1\)

4
Estimate \(\lim_{x \to 1^+} f(x).\)
\(x\)1.11.011.001
\(f(x)\)121021002
Show Answer

The outputs 12, 102, 1002 grow without a ceiling.

\(\lim_{x \to 1^+} f(x) = \infty\)

Answer: \(\infty\)

5
Estimate \(\lim_{x \to 1^-} f(x).\)
\(x\)0.90.990.999
\(f(x)\)\(-8\)\(-98\)\(-998\)
Show Answer

The outputs \(-8\), \(-98\), \(-998\) grow downward without a floor.

\(\lim_{x \to 1^-} f(x) = -\infty\)

Answer: \(-\infty\)

Level 2

Medium

Decide whether the two-sided limit exists, and whether the table fits continuity.

Example
The table gives values of f, and \(f(2) = 0\). Which statement is true?
\(x\)1.91.991.9992.0012.012.1
\(f(x)\)7.27.027.0026.9986.986.8
A. \(\lim_{x \to 2} f(x) = 0\)
B. \(\lim_{x \to 2} f(x) = 7\)
C. \(\lim_{x \to 2} f(x)\) does not exist
D. f is continuous at \(x = 2\)

Both sides approach 7, so the best estimate of the limit is 7.

\(f(2) = 0 \neq 7\)

f is not continuous at 2.

Answer: B. the limit is 7

1
Which statement is true?
\(x\)2.92.992.9993.0013.013.1
\(f(x)\)5.15.015.0011.0021.021.2
A. \(\lim_{x \to 3} f(x) = 5\)
B. \(\lim_{x \to 3} f(x) = 1\)
C. \(\lim_{x \to 3} f(x)\) does not exist
D. \(\lim_{x \to 3} f(x) = 3\)
Show Answer

\(\lim_{x \to 3^-} f(x) = 5\)

\(\lim_{x \to 3^+} f(x) = 1\)

The one-sided limits are not equal.

The two-sided limit does not exist.

Answer: C. the two-sided limit does not exist

2
Estimate \(\lim_{x \to 5} g(x).\)
\(x\)4.94.995.015.1
\(g(x)\)2.12.011.991.9
Show Answer

Left-hand values 2.1 and 2.01 approach 2.

Right-hand values 1.99 and 1.9 approach 2.

\(\lim_{x \to 5} g(x) = 2\)

Answer: \(2\)

3
The table gives values of f. Which statement is true?
\(x\)1.91.9922.012.1
\(f(x)\)4.24.0293.983.8
A. \(\lim_{x \to 2} f(x) = 9\)
B. the limit is 4, and f is not continuous at \(x = 2\)
C. \(\lim_{x \to 2} f(x)\) does not exist
D. f is continuous at \(x = 2\)
Show Answer

Nearby values 4.2, 4.02, 3.98, 3.8 approach 4.

\(\lim_{x \to 2} f(x) = 4\)

The table lists \(f(2) = 9\).

\(4 \neq 9\), so f is not continuous at 2.

Answer: B. the limit is 4, and f is not continuous at 2

4
Estimate \(\lim_{x \to 0} h(x).\)
\(x\)\(-0.1\)\(-0.01\)\(-0.001\)0.0010.010.1
\(h(x)\)0.010.00010.0000010.0000010.00010.01
Show Answer

From both sides, the outputs get close to 0.

\(\lim_{x \to 0} h(x) = 0\)

Answer: \(0\)

5
The table shows values of h near 0, and \(h(0)\) is not listed. Can you conclude that h is continuous at 0?
\(x\)\(-0.1\)\(-0.01\)\(-0.001\)0.0010.010.1
\(h(x)\)0.010.00010.0000010.0000010.00010.01
Show Answer

The table supports \(\lim_{x \to 0} h(x) = 0\).

Continuity also needs \(h(0)\) to equal that limit.

\(h(0)\) is not given.

The table does not show continuity.

Answer: no

Level 3

Hard

Two functions, or a conclusion a table cannot support.

Example
A classmate looks at the table and says the limit as x approaches 3 is about 3, because the average of 5.1 and 1.2 is 3.15. What is \(\lim_{x \to 3} f(x)?\)
\(x\)2.92.992.9993.0013.013.1
\(f(x)\)5.15.015.0011.0021.021.2

\(\lim_{x \to 3^-} f(x) = 5\)

\(\lim_{x \to 3^+} f(x) = 1\)

The one-sided limits are not equal.

Averaging a left-hand value and a right-hand value does not produce the two-sided limit.

The two-sided limit does not exist.

Answer: does not exist

1
Use the table. Estimate \(\lim_{x \to 2} \bigl(f(x)\bigr)^2.\)
\(x\)1.91.991.9992.0012.012.1
\(f(x)\)7.27.027.0026.9986.986.8

\(f(2) = 0\).

Show Answer

\(\lim_{x \to 2} f(x) = 7\)

\(\lim_{x \to 2} \bigl(f(x)\bigr)^2 = 7^2\)

\(49\)

\(\bigl(f(2)\bigr)^2 = 0\) is not the limit.

Answer: \(49\)

2
Use the table. Find \(\lim_{x \to 2} \bigl(f(x) + g(x)\bigr).\)
\(x\)1.91.992.012.1
\(f(x)\)6.36.035.975.7
\(g(x)\)2.42.047.967.6
Show Answer

f approaches 6 from both sides.

\(\lim_{x \to 2} f(x) = 6\)

g approaches 2 from the left and 8 from the right.

\(\lim_{x \to 2^-} g(x) = 2\)

\(\lim_{x \to 2^+} g(x) = 8\)

Left-hand limit of the sum:

\(6 + 2 = 8\)

Right-hand limit of the sum:

\(6 + 8 = 14\)

The one-sided limits differ, so the two-sided limit does not exist.

Answer: does not exist

3
Use the table above for f and g. Which statement is true?
\(x\)1.91.992.012.1
\(f(x)\)6.36.035.975.7
\(g(x)\)2.42.047.967.6
A. \(\lim_{x \to 2} g(x) = 5\)
B. \(\lim_{x \to 2} f(x)\) does not exist
C. \(\lim_{x \to 2} g(x)\) does not exist
D. \(\lim_{x \to 2} \bigl(f(x) + g(x)\bigr) = 11\)
Show Answer

f approaches 6 from both sides, so the limit of f exists.

g approaches 2 from the left and 8 from the right.

The limit of g does not exist.

5 would be an average of the two sides. That is not a limit.

The sum does not approach 11.

Answer: C. the limit of g does not exist

4
Which description fits the table?
\(x\)1.11.011.001
\(f(x)\)121021002
A. \(\lim_{x \to 1^+} f(x) = 0\)
B. \(\lim_{x \to 1^+} f(x) = 1002\)
C. \(\lim_{x \to 1^+} f(x) = \infty\)
D. the limit is a finite number near 1002
Show Answer

Each closer x-value produces a much larger output.

The outputs are not settling near one finite number.

\(\lim_{x \to 1^+} f(x) = \infty\)

Answer: C. the right-hand limit is \(\infty\)

5
Values of p approach 4 from both sides of \(x = 6\), and \(p(6)\) is not in the table. A student says p is continuous at 6 because the limit exists. What else is required?
Show Answer

A two-sided limit is one requirement for continuity.

Continuity at 6 also needs \(p(6)\) to be defined.

And it needs \(p(6) = \lim_{x \to 6} p(x)\).

The table does not give \(p(6)\), so continuity is not settled.

Answer: p(6) must equal the limit

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