Infinite Limits---The $$\frac n 0$$ Form

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Quick Overview

Suppose $$\red n\neq 0$$. Then...

  • If $$\displaystyle\lim\limits_{x\to a} f(x) = \frac{ \red n}{0} $$, the limit does not exist.
  • The $$\frac {\red n}{0}$$ form tells us the function is becoming infinitely large.
  • If the one-sided limits are the same, we say
    $$\displaystyle\lim\limits_{x\to a} f(x) = \infty\qquad\mbox{or}\qquad\lim\limits_{x\to a} f(x) = -\infty$$.

Notation

Examples

Example 1

Evaluate $$\displaystyle \lim_{x\to0}\,\frac 1 {x^2}$$

Step 1

Determine the form of the limit.

$$ \lim_{x\to0}\,\frac 1 {x^2} % = \frac{\red 1}{0} \qquad \left(\frac{\red n}{0} \mbox{ form}\right) $$

Since the limit has the $$\frac{\red n}{0}$$ form, we know the limit does not exist. However, it still might be an infinite limit.

Step 2

Examine the left-hand limit.

  • The numerator is positive.
  • Since the denominator is $$x^2$$, it will be positive.
  • The $$\frac{\red n}{0}$$ form tells us the function is becoming infinitely large.

Taken together, these three statements tell us $$\displaystyle \lim_{x\to0^-}\,\frac 1 {x^2} = \infty$$

Step 3

Examine the right-hand limit.

  • The numerator is positive.
  • Since the denominator is $$x^2$$, it will be positive.
  • The $$\frac{\red n}{0}$$ form tells us the function is becoming infinitely large.

Taken together, these three statements tell us $$\displaystyle \lim_{x\to0^+}\,\frac 1 {x^2} = \infty$$

Step 4

Conclusion

Both one-sided limits tell us the function is growing infinitely large in the positive direction. Since the one-sided limits indicate the same behavior, the limit is infinite.

Answer: $$\displaystyle \lim_{x\to0}\,\frac 1 {x^2} = \infty$$

Example 2

Evaluate: $$\displaystyle \lim_{x\to 3}\,\frac 1 {x-3}$$

Step 1

Determine the form of the limit.

$$\displaystyle\lim_{x\to 3}\,\frac 1 {x-3} = \frac 1 {3-3} = \frac{\red 1}{0}$$

The limit does not exist since it has the $$\frac{\red n}{0}$$ form. It might also be an infinite limit.

Step 2

Examine the left-hand limit.

  1. The numerator is always positive.
  2. The denominator will be negative, since $$x<3$$.
  3. The function is becoming infinitely large.

Taken together, these statements tell us $$\displaystyle \lim_{x\to 3^-}\,\frac 1 {x-3} = -\infty$$

Step 3

Examine the right-hand limit.

  1. The numerator is always positive.
  2. The denominator will be positive, since $$x>3$$.
  3. The function is becoming infinitely large.

Taken together, these statements tell us $$\displaystyle \lim_{x\to 3^+}\,\frac 1 {x-3} = \infty$$

Step 4

Conclusion

Since the one-sided limits are different, the limit does not exist.

Answer: $$\displaystyle \lim_{x\to 3}\,\frac 1 {x-3}$$ does not exist.

Example 3

Evaluate: $$\displaystyle \lim_{x\to2}\,\frac{x-5}{x^2-4x+4}$$

Step 1

Determine the form of the limit.

$$ \displaystyle\lim_{x\to2}\,\frac{x-5}{x^2-4x+4} % = \frac{2-5}{(2)^2-4(2)+4} % = \frac{-3}{(2)^2-4(2)+4} % = \frac{\red { -3}}{0} $$

The limit does not exist, but it might be an infinite limit.

Step 2

Factor the denominator to make our analysis easier.

$$ \displaystyle\lim_{x\to2}\,\frac{x-5}{x^2-4x+4} % = \displaystyle\lim_{x\to2}\,\frac{x-5}{(x-2)^2} $$

Step 3

Examine the one-sided limits.

  1. For both limits, the numerator will be -3.
  2. Since the denominator is being squared, it will always be positive.
  3. The function becomes infinitely large as $$x$$ approaches 2.

Since the numerator and denominator have opposite signs, the function will grow infinitely large in the negative direction.

Answer: $$\displaystyle \lim_{x\to2}\,\frac{x-5}{x^2-4x+4} = -\infty$$

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