Subsections

• Definition
• Exercises

Limits of functions.

Limits of many functions and expressions can be computed in Maple with the limit command.

  > limit(sin(x)/x,x=0);

  > g := x -> (3*x^2+4)/(x^2+7*x+12) ;

  > limit(g(x),x=-2);

  > limit(g(x),x=-4);

If the limit exists, Maple can usually find it. In cases where the limit doesn't exist, Maple gives the answer $\infty, -\infty$ or undefined for an unbounded limit or gives a range such as -1..1 if the limit doesn't exist, but the expression or function oscillates within these bounds.

  > limit(1/x,x=0);

  > limit(1/x^2,x=0);

  > limit(sin(1/x),x=0);

Maple can also do one-sided limits. Use Maple's online help to find out how to handle such limits.

The following definition of limit is given on page 72 of the text.

Definition

To say that $$\lim_{x\rightarrow c} f(x) = L$$ means that for each $\epsilon \gt 0$ (no matter how small), there is a corresponding $\delta \gt 0$ such that $\mid f(x) - L\mid < \epsilon$ provided that $0 < \mid x - c \mid < \delta$.

This definition may seem complicated, but it has a nice graphical interpretation. Plot f with the y-range set to $(L-\epsilon, L+\epsilon)$ and the x-range set to $(c-\delta, c+\delta)$. Try to choose $\delta$ small enough so that the graph of f stays between the top and bottom of the plot. If you can find such a $\delta$ no matter how small the $\epsilon$, then the limit exists and is equal to L.

Consider this example. Let $f(x) = x^2,\;\;c = 2$ and $\epsilon = 0.2$. Then any value for $\delta$ smaller than about 0.049 will work. To see why this is so look at the plots generated by the following commands.

  > f:=x->x^2;

  > limit(f(x), x = 2);

  > plot({4-0.2,4+0.2,f(x)},x=2-0.1..2+0.1,y=4-0.2..4+0.2);

  > plot({4-0.2,4+0.2,f(x)},x=2-0.048..2+0.048,y=4-0.2..4+0.2);

In the first of the two plot commands, the value of $\delta$ is 0.1. This is too large since the graph intersects the lines y = 4 - 0.2 and y = 4 + 0.2. The 0.048 value for $\delta$ in the second plot command, however, is small enough since the graph of f goes out the sides of the plot. Make sure you understand what is done in this example since you will need to do similar work in some of the exercises.

Exercises

In Exercises 1 and 2, use the limit command to find $$\lim_{x \rightarrow c}f(x)$$. Then experiment with some plots to find a $\delta$ that works for $\epsilon = 0.1.$ Also, find a $\delta$ that doesn't work. These two $\delta$ values should be close to each other. Thus your $\delta$ that works should be near the largest value of $\delta$ that can be used. Although, you will probably need to do many plots, submit only the two required plots with your worksheet. These plots should be in the style of the example above. Next, repeat the problem with $\epsilon = 0.01$.Conclude with a one paragraph statement of what you have learned in the exercise.

1.

$s(x) = \displaystyle\frac{x^3 + 2x^2 - x - 2}{x^3 - 6x^2 + 10x - 5}, \;\;c = 1$

2.

$w(x) = \displaystyle\frac{\sqrt{5+x^2} - \sqrt{5 - x^2}}{x^2},\;\;c = 0$

3.

Consider the following limit statement

If $\epsilon = 0.01$, find an appropriate $\delta$. Repeat your procedure with $\epsilon = 0.005$. What conclusion do you draw from your work? Why?

4.

Consider the following limit.

$$\lim_{x \rightarrow \frac{\pi}{2}} \tan(x) - \sec(x)$$

Use Maple to compute this limit. Then explain why this does not imply the existence of the two following limits.

$$\lim_{x \rightarrow \frac{\pi}{2}} \tan(x)$$

and

$$\lim_{x \rightarrow \frac{\pi}{2}} \sec(x)$$

(Hint: Does either of these limits exist?)