How to Solve One-Sided Limits

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  • One-sided limits are the same as normal limits, we just restrict $$x$$ so that it approaches from just one side
  • $$x\to a^+$$ means $$x$$ is approaching from the right.
  • $$x\to a^-$$ means $$x$$ is approaching from the left.

Examples

Example 1

Use the graph to approximate the value of both one-sided limits as $$x$$ approaches 3.

Step 1

Examine what happens as $$x$$ approaches from the left.

As $$x$$ approaches 3 from the left, the function seems to be approaching 2.

Step 2

Examine what happens as x approaches from the right

As $$x$$ approaches 3 from the right, the function seems to be approaching 3.

Answer Left-hand limit: $$\displaystyle \lim_{x\to3^-} f(x) \approx 2$$
Right-hand limit: $$\displaystyle \lim_{x\to3^+} f(x) \approx 3$$
Example 2: Using Tables

Using the tables below, what can be said about the one-sided limits as $$x$$ approaches 6?

$$ \begin{array}{l|c} x & f(x)\\\hline \hline 5 & 8.32571\\\hline 5.5 & 8.95692\\\hline 5.9 & 8.99084\\\hline 5.99 & 8.99987\\\hline 5.999 & 8.99992\\\hline 5.9999 & 8.99999\\\hline \end{array} $$

$$ \begin{array}{l|c} x & f(x)\\\hline \hline 7 & 1\\\hline 6.5 & 95\\\hline 6.1 & 1230\\\hline 6.01 & 9658\\\hline 6.001 & 54231\\\hline 6.0001 & 834366 \end{array} $$

Step 1

Examine what happens as $$x$$ approaches 6 from the left.

As $$x$$ approaches 6 from the left...

$$ \begin{array}{l|c} x & f(x)\\\hline \hline 5 & 8.32571\\\hline 5.5 & 8.95692\\\hline 5.9 & 8.99084\\\hline 5.99 & 8.99987\\\hline 5.999 & 8.99992\\\hline 5.9999 & 8.99999 \end{array} $$

...the function seems to be getting closer to 9.

Step 2

Examine what happens as $$x$$ approaches 6 from the right.

As $$x$$ approaches 6 from the left...

$$ \begin{array}{l|c} x & f(x)\\\hline \hline 7 & 1\\\hline 6.5 & 95\\\hline 6.1 & 1230\\\hline 6.01 & 9658\\\hline 6.001 & 54231\\\hline 6.0001 & 834366 \end{array} $$

...the function seems to just keep getting bigger.

Answer
$$\displaystyle\lim\limits_{x\to 6^-} f(x) \approx 9$$
$$\displaystyle\lim\limits_{x\to 6^+} f(x)$$ does not exist.

Practice Problems

Use the graph below to find the limits in questions 1--4.

Problem 1

$$\displaystyle \lim_{x\to-4^-} f(x) \approx$$

Answer: $$\displaystyle\lim\limits_{x\to-4^-} f(x)$$ does not exist.
Problem 2

$$\displaystyle \lim_{x\to-4^+} f(x) \approx$$

Answer: $$\displaystyle\lim\limits_{x\to-4^+} f(x) \approx -2$$
Problem 3

$$\displaystyle \lim_{x\to 1^-} f(x) \approx$$

Answer: $$\displaystyle\lim\limits_{x\to 1^-} f(x) \approx -3$$
Problem 4

$$\displaystyle \lim_{x\to 1^+} f(x)\approx$$

Answer: $$\displaystyle\lim\limits_{x\to1^+} f(x) \approx 1$$
Problem 5

Use the tables below to evaluate both one-sided limits as $$x$$ approaches -4.

$$ \begin{array}{l|c} x & f(x)\\\hline \hline -5 & -4.5\\\hline -4.5 & -44.5\\\hline -4.1 & -444.5\\\hline -4.01 & -4444.5\\\hline -4.001 & -44444.5\\\hline -4.0001 & -444444.5 \end{array} $$

$$ \begin{array}{l|c} x & f(x)\\\hline \hline -3 & 2.13671\\\hline -3.5 & 2.52240\\\hline -3.9 & 2.59684\\\hline -3.99 & 2.59987\\\hline -3.999 & 2.59990\\\hline -3.9999 & 2.59999 \end{array} $$

Step 1

Examine what happens as $$x$$ approaches -4 from the left.

As $$x$$ approaches -4 from the left...

$$ \begin{array}{l|c} x & f(x)\\\hline \hline -5 & -4.5\\\hline -4.5 & -44.5\\\hline -4.1 & -444.5\\\hline -4.01 & -4444.5\\\hline -4.001 & -44444.5\\\hline -4.0001 & -444444.5 \end{array} $$

...the function just keeps getting bigger.

Step 2

Examine what happens as $$x$$ approaches -4 from the right.

As $$x$$ approaches -4 from the right...

$$ \begin{array}{l|c} x & f(x)\\\hline \hline -3 & 2.13671\\\hline -3.5 & 2.52240\\\hline -3.9 & 2.59684\\\hline -3.99 & 2.59987\\\hline -3.999 & 2.59990\\\hline -3.9999 & 2.59999 \end{array} $$

...the function seems to get closer to 2.6.

Step 4
Left-hand limit: $$\displaystyle\lim\limits_{x\to -4} f(x)$$ does not exist. Right-hand limit: $$\displaystyle\lim\limits_{x\to-4} f(x) \approx$$ 2.6.
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