Limits from table of values

Make an educated guess - IB, probably

Contents

• Why limits
• Table of values
• Two-sided limits
• One-sided limits
• Can only approach from one side
• Limit does not exist

Why limits

There are many practical uses of limits in this course.

1. Allow easier curve sketching around vertical, horizontal, and even slant asymptotes
2. Check answers in calculus and non-calculus contexts by seeing if it makes sense with extreme values
3. Help to answer questions regarding range
4. Question asks about limits and it’s nice to have some easy marks

Table of values

SL questions only require finding a limit using a table of values. For $\displaystyle \lim_{x\to 0}\frac{\sin x}{x}$, the table may look like

Then conclude $\displaystyle \lim_{x\to 0}\frac{\sin x}{x} = 1$. If you are getting $0.01745$, then set your calculator to radians mode.

On a calculator this is done using tables, lists or spreadsheets.

Two-sided limits

The above limit was a two sided limits approaching $0$ from the left and the right. If both directions did not approach the same value, then the limit does not exist.

One-sided limits

An one-sided limit approaches from only the left ($^-$) or the right ($^+$), eg

Can only approach from one side

Consider continuous $g(x)$ defined on $x\in\left[1,3\right[$, limit as $x$ approaches $3$ is only possible from the left, as $x > 3$ is not in the domain. Then $\displaystyle {\lim_{x\to 3}g(x)}$ is defined to be $\displaystyle {\lim_{x\to 3^-}g(x)}$.

The most common way this appears is as $x$ approaches $\pm \infty$. Eg $\displaystyle {\lim_{x\to -\infty} \e^x = 0}$, without having to write $\displaystyle {\lim_{x\to -\infty^+} \e^x = 0}$

Limit does not exist

A limit does not exist if any of the following is true:

• the limit “approaches” $-\infty$ or $\infty$. ie divergence means no limit, OR
• the two-sided limit does not exist when the two one-sided limit disagree.