Limit Does Not Exist: Why and How in Simple Steps


When Does a Limit Not Exist?

In order for a limit to exist at a point c, it must settle on a certain value at that point. There are three main reasons for a limit not existing:

  1. Wild oscillations: the function bounces between two x-values as x approaches c,
  2. The function settles on two different numbers: one approaching from the left and a different one from the right as x approaches c.
  3. unbounded behavior: the function increases or decreases without constraint as x approaches c.

1. Wild Oscillations


graph of a function where the limit does not exist
The limit does not exist at x = 0 for the sine function y = x sin(1/x).

What’s important is that the values around a point settle towards a number, not that a particular function value exists. A case in point is y = x sin(1/x). Although the function value at x = 0 exists (it’s 0), the function varies wildly around that point. For example, testing a few values around 0, you get:

  • y = .5 sin(1/.5) = 0.45
  • y = 1 sin(1/1) = 0.84

Therefore, the limit doesn’t exist for y = x sin(1/x). However, you can make a good approximation with the Squeeze Theorem.

Function settles on two different numbers

The function settles on more than one number as you move in towards your chosen x-value from the left and right.

step function limit
This graph of y = |x|/x does not have a limit at x = 0 because left and right approaches have different numbers (1 and -1).

Unbounded behavior

Unbounded behavior of a limit refers to a function growing without bound (in other words, to infinity) at the limit point. The following image shows unbounded behavior near zero, where the y-values increase without bound:
unbounded limit doesn't exist

How to Find When the Limit Does Not Exist

There are several ways to find out if a limit does not exist. From easiest to more challenging, they are:

  1. Limit calculator
  2. Graph the function
  3. Create a table of values
  4. Use algebra

1. Limit Calculator

This Wolfram widget will calculate the limit for you:

2. Graphing

Look at the graph. If the graph is going in completely different directions (i.e. up and down at the same point) at the particular x-value you are trying to find a limit for, the limit does not exist.

graph of limit does not exist
The graph is going in opposite directions at x=2, so the limit does not exist at that point.

On a graphing calculator, zoom in on smaller and smaller increments to test the behavior of the graph (See: How to use zoom on the TI-89).

3. Using a Table

By hand, the easiest way to show a limit doesn’t exist is to calculate the one-sided limits; In other words, find the limit as it comes from the left, and find it as it comes from the right. For example, let’s say you have some function f(x) = [x] at x = 2, then find that:

  • lim (x→2−) = 1
  • lim (x→2+) = 2

Then the limit for f(x) does not exist.

Find the limit of the function on a TI-89 by building a table with small increments either side of the function’s value. For example, if you want to know if the limit exists at x = 1, then make your inputs several values around x = 1, like {0.9, 0.99, 1. 01, 1.1 }. If the table shows a trend to different numbers either side of the value, then the limit does not exist. See: How to Build a Table of Values on the TI-89.

4. Using Algebra to find when the limit does not exist

Use your algebra skills to look at the behavior of a function. For example, the limit of the function
y = (-1/(1-x)-cos(x))/(2cos(x)sin(x))
does not exist at x = 0. As the function approaches zero, the numerator approaches -2 and the denominator approaches 0. Using this logic, you can determine that the limit from the left is negative infinity and the limit from the right is infinity.

Tip: Technically, a limit doesn’t exist if the value at that function is infinity. But knowing that a number approaches infinity at a certain point is extremely useful, so we say that: lim f(x) = ∞.

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How to Use Zoom on the TI-89

  1. Press ON.
  2. Press the green button, then the F1 button at the top to access the ‘Y=’ command.
  3. Type in your function.
  4. Press the F2 button for ZOOM options.
  5. set factors zoom in ti89

    Choose Zoom in. This recenters the graph, zooming in by the amount you specify in C: SetFactors. Use the arrows to scroll down to this setting. You may not need to change the settings; The default is 1/4 of the x-axis and y-axis (your scales will be 1/4 of what they were previously).

Limit Does Not Exist: References

Folk, D. (Undated). Graphing a Function on the TI-89. Retrieved May 25, 2019 from:
Larson, R. & Edwards, B. (2009). Calculus. Cengage Learning.

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