Limits from Both Sides

The four graphs come from the Lesson 26 Notes and slides. On each graph, a filled dot marks a function value and an open dot marks a point missing from the graph. The three formulas are

$$\frac{\sin(x)}{x}, \qquad \frac{1 - e^x}{x}, \qquad \frac{|x|}{x}.$$

The dashed blue line marks $x = a$. The orange points approach that line from the left and the right. Shrink ÷10 moves both points ten times closer. Press Right, Left, or Both to choose the sides shown.

Show numbers gives the outputs at the orange points and a table of $f(a - d)$ and $f(a + d)$ as the distance $d$ shrinks. Once the third decimal on one side stops changing, the worked column gives that side’s estimate of its one-sided limit:

$$\lim_{x \to a^-} f(x) \quad \text{and} \quad \lim_{x \to a^+} f(x).$$

The limit exists exactly when both one-sided limits exist and are equal. The worked column ends with $f(a)$ and says whether it equals the limit. Click a table row to place the points at that row’s distance.

The + and − buttons zoom in on $a$ and back out. Fit points zooms until both points fill the view. With Follow on, the view zooms in with the points as they move toward $a$.

Under More, type a function as in R and press Graph to draw it.

Show R gives the R code that defines the function with makeFun() and evaluates it at the points and at $a$. For the four graphs, Show R gives no code, because a graph has no formula.

Keys: press L, R, or A to pick the left point, the right point, or the line at $a$. Press [ or ] to step through the choices. Press the left or right arrow key to move it one step. Hold Shift with an arrow key to move it five steps. Press Esc to let go.