How to Detrmine when Limits Don't Exist

When the limit does not exist, as an animated GIF

when calculus limit does not exist animated gif More calculus gifs

Quick Summary

Limits typically fail to exist for one of four reasons:

  1. The one-sided limits are not equal
  2. The function doesn't approach a finite value (see Basic Definition of Limit).
  3. The function doesn't approach a particular value (oscillation).
  4. The $$x$$ - value is approaching the endpoint of a closed interval

Examples

Example 1: One-sided limits are not equal

Use the graph below to understand why $$\displaystyle\lim\limits_{x\to3} f(x)$$ does not exist.

$$f(x)$$ approaches two different values...

...depending on which direction $$x$$ approaches from.

In the graph, we notice that $$\displaystyle\lim_{x\to3^-} f(x) \approx 2$$ and $$\displaystyle\lim_{x\to3^+} f(x) \approx 3$$

Even though the graph only allows us to approximate the one-sided limits, it is certain that the value $$f(x)$$ is approaching depends on the direction $$x$$ is coming from. Therefore, the limit does not exist.

Example 2: Infinitely Large Value

Use the graph below to understand why $$\displaystyle\lim\limits_{x\to 3} f(x)$$ does not exist.

In order for a limit to exist, the function has to approach a particular value. In the case shown above, the arrows on the function indicate that the the function becomes infinitely large. Since the function doesn't approach a particular value, the limit does not exist.

Example 3: Infinite Oscillations

What is $$\displaystyle\lim\limits_{x\to0} \sin(\frac 1 x)$$?

Something interesting happens when you examine $$f(x) = \sin\left(\frac 1 x\right)$$ as $$x$$ approaches 0. The function begins to oscillate faster and faster.

The closer $$x$$ gets to 0, the faster the function oscillates between 1 and -1. Is $$f(x)$$ approaching a single, particular value? No, it isn't. Consequently, the limit does not exist.

Example 4: End Points of an Interval

Examine $$\lim\limits_{x\to0} \sqrt x$$

Consider the graph of $$f(x) = \sqrt x$$ below. How would we determine the limit as $$x$$ approaches 0?

Since this function is only defined for $$x$$-values to the right of 0, we can't let $$x$$ approach from the left.

In order to say the limit exists, the function has to approach the same value regardless of which direction $$x$$ comes from (We have referred to this as direction independence). Since that isn't true for this function as $$x$$ approaches 0, the limit does not exist.

In cases like thi, we might consider using one-sided limits.

back to One-Sided Limits next to Limit Laws

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