Math 124: Limits at infinity & Horizontal Asymptotes

Tolstoy, Count Lev Nikolgevich (1828-1920)
A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself.
The larger the denominator the smaller the fraction.
In H. Eves Return to Mathematical Circles, Boston: Prindle, Weber and Schmidt, 1989.

1. Convince yourself that .

You can use the graph, or just realize that you can get 1/x as close to zero as you want by just picking a large enough (positive or negative) value for x. For example, you can get

by taking

Also: : For example, you can get

by taking

More generally: , i.e. limits at infinity of any negative powers of x are zero.

Also:

And if, then

(On the other hand, for positive powers a, and , depending (on what?) )

2. A most useful technique in computing limits at infinity when the expression inside the limit is a fraction involving powers of x, is to divide each term in both the numerator (top) and denumerator (bottom) of the fraction by an appropriate power of x.

This is because we have a limit of the type (undefined), so depending on the particular case, the limit might end up being a finite number (zero or non-zero), +∞, or –∞.

Example:

:

Example:

Here,

Example:

This limit is of the type ∞-∞ (also undefined).
Here you need to rationalize first, then use the above technique:

Now divide by x both top and bottom:

3. Sometimes you just need to know your functions:

Example: does not exist,

because cos(Ө) keeps cycling through all values between -1 and 1 as Ө gets smaller and smaller towards -∞.

(on the other hand, . Why? Because x does not depend on Ө, so for the purposes of that limit we can treat cos(x) as a constant and apply the Constant Law)

Example:

Example:

4. Find all the asymptotes for the function

a) Horizontal asymptotes are the lines y=L such that . So we need to check the limits of this function at +∞ and –∞.

Similarly, check that .

So this function has the same horizontal asymptote, y=1, at both (could be different!)

b) Vertical asymptotes are the lines x=a such that .

NOTE THE DIFFERENCE BETWEEN THE TWO TYPES OF ASYMPTOTES!

This function can only have infinite limits at the zeroes of the denominator. But not necessarily (why not?). So we need to check carefully:

First, factor the denominator:. So the values of x to check are x=3 and x=-1:

· .

Hence x=3 is not a vertical asymptote (rather, the function has a removable discontinuity at x=3)

· does not exist, because the limit to the left is & the limit to the right is :

Hence x=-1 is a vertical asymptote

Conclusion:

· this function is defined for all x, except x=3 and x=-1.

· It has one horizontal asymptote: y=1, at both

· It has one vertical asymptote: x= -1.

Cool!

Computing rates of change (velocities)

Recall our motivation for limits: to compute instant rates of change!

Rate of Change	See on graph	Compute as	Example
Average rate of change of f(x) between x=a and x=a+h	Slope of secant line through the graph of f(x) at x=a and x=a+h		Average speed: If you did 10 miles in 10 minutes, your average speed was 1 mpm (= 60mph)
Instantaneous rate of change of f(x) at x=a	Slope of tangent line to the graph of f at x=a		Actual speed: If get a speeding ticket, your speed at the instant you pass the radar gun was faster than the legal limit.

Example 1: The following is the graph of the distance travelled by a car after t minutes.

When was its speed zero? Was it driving faster at t=2 or at t=3?

What can you say about its speed pattern?

Answers:

The speed is zero when the tangents have slope zero (are horizontal). That is, at t=4.5 and t=9, approximately.

The slope of the tangent line is steeper at t=2 than at t=3. That is, the car is driving faster at t=2 than at t=3.

Follow the slopes of the tangents along the graph (use a ruler if it helps you visualize it). The speed starts off large and positive at t=0, decreases to zero at about t=4.5, becomes negative from t=4.5 to t=9 (first decreasing up to about t=6.5, then increasing towards zero), then becomes positive and increasing again.

Example 2: Let d(t)=t²+2.

a) What is the slope of the tangent line at t=3?

Compute the slope of the secant line:

=

Then take the limit as h->0 of the resulting expression in h:

The slope of the tangent line is

b) What is the formula for the speed s(t)?

For an arbitrary time t, compute the average velocity over a time interval of h minutes starting at t:

Then take the limit as h->0 and get a function in t: s(t)=2t.