Recognize when a limit law is inconclusive

Three quotients with the same problem

The three curves are quotients: \(A = \tfrac{3x}{x}\), \(B = \tfrac{x^2}{x}\) and \(C = \tfrac{x}{x^2}\). For each one, the top and the bottom approach 0 as \(x\) approaches 0.

The orange dots mark the height of each curve at the input \(x\). Move \(x\) toward 0 from the right. The height of A stays at 3, and the height of B approaches 0. The height of C grows without bound, from 10 at \(x = 0.1\) to 100 at \(x = 0.01\), so its dot leaves the picture.

The quotients have the same form but different outcomes, and the quotient law cannot decide them because the bottom approaches 0.

A law needs its conditions

The quotient law has an "if" part and a "then" part. If both limits exist and the limit of the bottom is not 0, then the limit of the quotient is the quotient of the limits.

When the "if" part fails, the law is silent: it does not say whether the limit exists.

Show each curve alone. All three fail the "if" part in the same way, yet A and B have limits and C has none, so the quotient itself decides.

Notation

1. Suppose \(f(x)\) approaches \(\cy{L}\) and \(g(x)\) approaches \(\cy{M}\) as \(x\) approaches \(a\). If both limits exist and \(\cy{M}\) is not 0, the quotient law applies.

   \[\begin{aligned} \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\cy{L}}{\cy{M}} \\ \cy{M} \neq 0 \end{aligned}\]

2. If \(M = 0\) and \(L\) is not 0, the quotient is very large in size near \(a\), as for the curve \(\tfrac2x\). It has no limit, and its values show this, not the law.

3. If \(M = 0\) and \(L = 0\), the quotient can approach any number or have no limit, as A, B and C show. You change the expression to find out.

4. The sum law also needs both limits to exist. Two functions can have no limit while their sum has one.

   \[\begin{aligned} \lim_{x \to 0} \frac1x \ \text{does not exist} \\ \lim_{x \to 0} \Bigl(\frac1x + \Bigl(-\frac1x\Bigr)\Bigr) = 0 \end{aligned}\]

When it fails

Neither \(\tfrac1x\) nor \(-\tfrac1x\) has a limit at 0, so the sum law does not apply. The sum is 0 at every input except 0, so its limit is 0.

A failed condition proves nothing about the limit, because the law then gives no answer.

Example 1: the values decide

Find the limit of \(\dfrac{3x+1}{x-4}\) as \(x\) approaches 4.

1. The bottom approaches 0, so the quotient law is silent.

   \[\begin{aligned} \lim_{x \to 4} (3x+1) = 13 \\ \lim_{x \to 4} (x-4) = 0 \end{aligned}\]

2. The top approaches 13 while the bottom approaches 0, so the quotient is very large in size. To the right of 4 the bottom is positive, so the curve is green and rises without bound.

   \[\frac{3(4.01)+1}{4.01-4} = 1303\]

3. To the left of 4 the bottom is negative, so the curve is red and falls without bound. The two sides do not approach one number, so the limit does not exist.

   \[\frac{3(3.99)+1}{3.99-4} = -1297\]

Example 2: the law is silent, and the limit exists

Find the limit of \(\dfrac{x^2-25}{x-5}\) as \(x\) approaches 5.

1. The top and the bottom both approach 0, so the quotient law is silent.

   \[\begin{aligned} \lim_{x \to 5} (x^2-25) = 0 \\ \lim_{x \to 5} (x-5) = 0 \end{aligned}\]

2. A tempting wrong move: The law fails, so the limit does not exist.

   \[\lim_{x \to 5} \frac{x^2-25}{x-5} \ \text{does not exist}\]

   A silent law gives no answer in either direction. The dots approach height 10 from both sides of the open circle, so the limit exists.

3. Change the expression by factoring the top. You may cancel \(x-5\) because the limit uses only inputs other than 5.

   \[\frac{(x-5)(x+5)}{x-5} = x+5 \ \to \ \cy{10}\]

Practice

Find the limit of \(\dfrac{x^2+3x-4}{x+4}\) as \(x\) approaches \(-4\).

Move the slider toward −4 to see the height the dots approach. To find it exactly, look for a factor that the top and the bottom share.

Questions

The graph shows two quotients, \(P\) and \(Q\). An open circle marks the missing point of \(P\) at \(x = 1\), and the dashed line marks \(x = 3\), where \(Q\) has no value.

1. The bottom of \(P\) approaches 0 at \(x = 1\). What does the graph show about the limit of \(P\)?

   • The limit is 4
   • The limit does not exist
   • The limit is 0
   Answer

   The limit is 4

2. The top of \(Q\) approaches 5 and the bottom approaches 0 at \(x = 3\). What does the graph show about the limit of \(Q\)?

   • The limit does not exist
   • The limit is 5
   • The limit is 0
   Answer

   The limit does not exist

3. What is the limit of \(\dfrac{x^2-4}{x+2}\) as \(x\) approaches \(-2\)?

   • −4
   • 0
   • It does not exist
   • 4
   Answer

   −4

Summary

When a condition of a limit law fails, such as a bottom that approaches 0, the law gives no answer.

A law states only what follows when its conditions hold, so a failed condition leaves the limit undecided.

You then find the limit another way, by changing the expression or by reading the values.

This is one of four skills in topic 1.5 of Which & Why: AP Calculus.