Undefined Limits: Definition & Examples

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Some situations in math are undefined such as 1/0 and the square root of negative numbers. In this lesson, we will deal with scenarios where limits don't exist.

Error

Enter into your calculator the following problems:

(a) 1/0

(b) √-1

Your calculator should have returned the error message because these scenarios are not defined! Part (a) is a value of x in the function f(x) = 1/x where there is no finite y-value. The closer the value of x gets to 0 the y-value either approaches negative or positive infinity. Which infinity it approaches depends on which way you move along the x-axis. Part (b) is undefined because there is no number that multiplied by itself gives -1. There are some limits in Calculus that are undefined too. Let's look at the types of limits that are undefined.

One-Sided Limits

One-sided limits are the evaluation of a function as x approaches a value from the left side and right side of a function. If there are different y-values as x approaches from the left and right sides the limit doesn't exist. Let's look at the graph of a function where the one-sided limit doesn't exist. Graph 1 shows the function.


Graph 1
graph1


The type of limit for this graph may be formally written as

limit1

Looking back at Graph 1 we see that as we trace the function from the left heading towards x = 1 we end up at f(x) equaling 1. As we approach x = 1 from the right we end at a different value for f(x), which is 3. Since 1 can not equal 3, this limit is undefined! Now let's look at another type of undefined limit.

Infinite Oscillations

Some trigonometric functions start to oscillate so radically between two y-values as x approaches a specific value the limit is undefined. The graph of f(x) = cos(1/x) is shown in Graph 2.


Graph 2
ex2a


We can see from Graph 2 as x approaches 0 from either side the graph becomes like a compressed accordion shape that is oscillating in an extreme fashion, which is undefined limit in Calculus! This is called an infinite oscillator. Let's head towards the third type of undefined limit.

End Point Intervals

End point intervals are similar to one-sided limits because we approach an x-value from a specific direction. The difference is we can only approach a specific x-value from one direction. An example of this type of limit is a function of the position of an object with respect to time. The approach to time = 0 along the x-axis can only be done from positive values of x. We could never approach 0 from negative x-values because the position function doesn't exist at negative x-values. Time can never be a negative value! Graph 3 shows how there is no function at negative time values along the x-axis.


Graph 3
ex3


We have one more undefined limit to discuss.

No Finite Value Limits

If a function does not approach a finite value from either direction the limit is undefined. Graph 4 is a good way to see what is meant by a function not reaching a finite value.


Graph 4
Graph4


As x-values approach 1 from either side the y-value approaches positive infinity. Since infinity is not a finite value, the limit of the function as x approaches 1 is undefined. Let's now look at how to determine if a limit approaches a finite value if no graph is given.

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