← Back to learning objective

One-sided infinite limits at a vertical asymptote

Consider the function f(x)=1(x−3)3f(x) = \dfrac{1}{(x-3)^3}, which has a vertical asymptote at x=3x = 3.

One-sided infinite limits at a vertical asymptote graph

Using the graph above (or by reasoning directly about the formula) determine the following two one-sided limits:

lim⁡x→3−1(x−3)3andlim⁡x→3+1(x−3)3\lim_{x \to 3^-} \frac{1}{(x-3)^3} \qquad \text{and} \qquad \lim_{x \to 3^+} \frac{1}{(x-3)^3}

State the value (or +∞+\infty / −∞-\infty) of each one-sided limit.

Hint

Think about the sign of (x−3)3(x-3)^3 when xx is slightly less than 3 versus slightly greater than 3. Since the exponent 3 is odd, the sign of (x−3)3(x-3)^3 matches the sign of (x−3)(x-3), and cubing a small number keeps that sign while making the denominator's magnitude tiny (so the fraction's magnitude grows large).

Step-by-step solution

As x→3−x \to 3^-, the quantity x−3x-3 is a small negative number, so (x−3)3(x-3)^3 is also a small negative number (since cubing preserves sign). Dividing 1 by a very small negative number produces a number with very large magnitude and negative sign, so lim⁡x→3−1(x−3)3=−∞\lim_{x\to 3^-} \frac{1}{(x-3)^3} = -\infty.

As x→3+x \to 3^+, the quantity x−3x-3 is a small positive number, so (x−3)3(x-3)^3 is also a small positive number. Dividing 1 by a very small positive number produces a number with very large magnitude and positive sign, so lim⁡x→3+1(x−3)3=+∞\lim_{x\to 3^+} \frac{1}{(x-3)^3} = +\infty. This confirms x=3x=3 is a vertical asymptote with opposite infinite behavior on each side.

Sign of the infinite limit approaching from the left

For the function f(x)=1(x−3)3f(x) = \dfrac{1}{(x-3)^3}, approaching three from the left, does the function grow toward positive or negative infinity?

Hint
Step-by-step solution

The function grows toward negative infinity as x approaches three from the left. Just below three, x minus three is a small negative number, and cubing keeps that negative sign. Dividing one by a very small negative number gives a large negative result, matching the earlier finding that the left-hand limit equals negative infinity.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap