calculuspowerup.com

Limit Subtleties and Comments on Infinity

April 8th, 2009 |

Author: Merrin

In calculus, we are often concerned with limits at points that are a bit trickier than for ordinary spots in continuous functions. We might need to calculate limits of functions that are the ratio of two expressions that are both nearly zero or the ratio of very large numbers. We also might need to calculate a limit around a point where there is a hole in the function or even a discontinuous jump. Functions may also oscillate near the limiting value. Getting finite values out of these limits or determining that they do not exist, DNE, is part of calculus.

If an expression is not continuous at a point, try calculating the left and

right hand limits. One substitution which is helpful in calculating left and right hand limits is to always write them both in terms of a limit that approaches zero from the right. Hence from the right we can write

$\mathop{\lim }\limits_{x\to c^{+} } f(x)=\mathop{\lim }\limits_{\varepsilon \to 0^{+} } f(c+\varepsilon )$

And from the left we can write

$\mathop{\lim }\limits_{x\to c^{-} } f(x)=\mathop{\lim }\limits_{\varepsilon \to 0^{+} } f(c-\varepsilon )$

In these limits, epsilon is always a positive quantity, which helps get the sign correct in our expressions when function straddles zero or changes signs. When calculating limits of trickier situations sometimes the limit does not exist, or does not exist because it is infinity. It is worth a few words on the concept of infinity since we will encounter it in our calculations often. Infinity is not a number, it is a concept. Think of the biggest number possible. There is always a bigger number, for example just add one to your number. Infinity is this process repeated over and over again forever.

The expression 1/0 is undefined and is not infinity. Division by zero is not allowed and is undefined. The expression

$1/0^+$

is infinity. Think of dividing one by the smallest number imaginable. There is always a smaller number. On the same note,

$1/0^{-}$

(1 divided by the smallest possible negative number) is negative infinity.

When a limit approaches infinity then the limit technically does not exist because infinity is not a number. We can always be specific about why it doesn’t exist because it is more informative thanDNE and write.

$\mathop{\lim }\limits_{\varepsilon \to 0^{+} } \cfrac{1}{\varepsilon } =\infty$

This is said as, “The limit of one over epsilon as epsilon tends to zero from the right equals infinity.”

Indeterminate forms are expressions involving numbers and infinity that cannot be evaluated by inspection. Indeterminate forms might take any value. The form for zero times infinity can equal zero, a finite number, or infinity as shown in these three different examples.

$\left(\mathop{\lim }\limits_{x\to \infty } \cfrac{1}{\sqrt{x} } \right)\left(\mathop{\lim }\limits_{x\to \infty } x\right)~\mathop{\lim }\limits_{x\to \infty } \sqrt{x} =\infty \\\left(\mathop{\lim }\limits_{x\to \infty } \cfrac{1}{x^{2} } \right)\left(\mathop{\lim }\limits_{x\to \infty } x\right)~\mathop{\lim }\limits_{x\to \infty } x^{-1} =0 \\\left(\mathop{\lim }\limits_{x\to \infty } \cfrac{3}{x} \right)\left(\mathop{\lim }\limits_{x\to \infty } x\right)~\mathop{\lim }\limits_{x\to \infty } 3=3$

Now assuming you can go ahead and combine these limits you can make those calculations. In the first case, the quantity in the numerator is a more powerful infinity then the denominator and it wins out. In the second case, the opposite is

the case. In the last case, the powers balance out.

Indeterminate form can take different values for different examples. Some other tricky indeterminate forms are

$\infty -\infty \qquad 0^{0}\qquad\infty ^{0}\qquad 1^{\infty }$

We will later revisit these situations when we have more techniques built up to deal with limits.

Infinity pops up sometimes when we take certain limits of our elementary functions. Go through the following list and see that each limit makes sense. These limits are mainly stating what the behavior of a function after extending their graphs to infinity.

${\mathop{\lim }\limits_{x\to \infty } \left|P(x)\right|=\infty \qquad\deg (P)>0}$

${\mathop{\lim }\limits_{x\to 0+} \ln x=-\infty \qquad \mathop{\lim }\limits_{x\to \infty } \ln x=\infty }$

${\mathop{\lim }\limits_{x\to \infty } e^{x} =\infty \qquad \mathop{\lim }\limits_{x\to -\infty } e^{x} =0}$

${\mathop{\lim }\limits_{x\to \infty } a^{x} =\infty \quad a>1\qquad \mathop{\lim }\limits_{x\to -\infty } a^{-x} =0\quad a>1}$

${\mathop{\lim }\limits_{x\to \infty } \sin x=DNE\quad \quad \quad \quad \quad \mathop{\lim }\limits_{x\to \infty } \cos x=DNE}$

${\mathop{\lim }\limits_{x\to \infty } \tan x=DNE\quad \quad \quad \quad \quad \, \mathop{\lim }\limits_{x\to \infty } \sec x=DNE}$

${\mathop{\lim }\limits_{x\to \infty } \csc x=DNE\quad \quad \quad \quad \quad \mathop{\lim }\limits_{x\to \infty } \cot x=DNE}$

${\mathop{\lim }\limits_{x\to \infty } \cosh x=\mathop{\lim }\limits_{x\to \infty } \cfrac{e^{x} }{2} =\infty }$

$\mathop{\lim }\limits_{x\to \infty }\sinh x=\mathop{\lim }\limits_{x\to \infty }\cfrac{e^{x} }{2} =\infty$

$\mathop{\lim }\limits_{x\to \infty } \tanh x=1$

${\mathop{\lim }\limits_{x\to \infty } \left|P(x)/Q(x)\right|=0\quad \deg P<\deg Q}$

${\mathop{\lim }\limits_{x\to \infty } \left|P(x)/Q(x)\right|=a \quad\deg P=\deg Q}$

${\mathop{\lim }\limits_{x\to \infty } \left|P(x)/Q(x)\right|=\infty \,\,\,\,\,\deg P>\deg Q,}$

Think about each of these limits until they are intuitive. We now move on to some more particular examples of evaluating limits where we can’t use continuity.

Bookmark It

(No Ratings Yet)

Loading ...

Posted in Limits

calculuspowerup.com

Limit Subtleties and Comments on Infinity

Leave a Reply