Infinite Limits

What are infinite limits?

You may come across limits whose value is infinity or minus infinity. That is, as x approaches a certain point, the value of the function becomes significantly larger and larger. To further investigate infinite limits, let us take a look at the limit below:

[latex]\LARGE{\lim_{x\to 0}\frac{1}{x^2}}[/latex]

What happens to the function as x approaches zero? The table below shows the values of f(x) for different values for different values of x, approaching very close to 0.

It is clear that as x approaches 0, the value of f(x) approaches a very large number. In other words, we can keep increasing the value of x, and the value of the function f(x) will become larger and larger. Since the value is increasing without bound, we say that the value of f(x) approaches infinity in this case. We will denote this by saying: [latex]lim_{x\rightarrow 0}\frac{1}{x^2} = \infty[/latex]. The general notation for this is:

Notation:

[latex]\lim_{x\rightarrow a}f(x)= \infty[/latex]

[latex]\lim_{x\rightarrow a}f(x)= -\infty[/latex]

Example 3

For the function [latex]f(x) = \frac{1}{x} [/latex], find the following limits (if they exist):

(a) [latex]\lim_{x \to 0^-}f(x)[/latex]
(b) [latex]\lim_{x \to 0^+}f(x)[/latex]
(c) [latex]\lim_{x \to 0}f(x)[/latex]

Solution

Let us first graph the function [latex]f(x) = \frac{1}{x}[/latex], as shown below.

Part a:

From the graph, we can see that if we approach 0 from the left side of the function, the value of the function approaches a large value in the negative direction, and continues to do so as we keep approaching zero. Thus, it is clear that:

[latex]\Large{\lim_{x \to 0^-}(\frac{1}{x}) = -\infty}[/latex]

Part b:

Similarly, if we approach 0 from the right side, the value of the function approaches a large number in the positive direction, or, positive infinity. Thus:

[latex]\Large{\lim_{x\to 0^+}(\frac{1}{x}) = \infty}[/latex]

Part c:

We know that if the left hand limit and the right hand limit at a given point are different, then the overall limit at that point does not exist. To summarize this idea, since:

[latex]\Large{\lim_{x \to 0^-}(\frac{1}{x}) \neq \lim_{x \to 0^+}(\frac{1}{x}) }[/latex]

Then:

[latex]\Large{\lim_{x \to 0}(\frac{1}{x})}[/latex] does not exist!

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Solution

Let us first graph the function [latex]f(x) = \frac{1}{x}[/latex], as shown below.

Part a:

[latex]\Large{\lim_{x \to 0^-}(\frac{1}{x}) = -\infty}[/latex]

Part b:

Similarly, if we approach 0 from the right side, the value of the function approaches a large number in the positive direction, or, positive infinity. Thus:

[latex]\large{\lim_{x\to 0^+}(\frac{1}{x}) = \infty}[/latex]

Part c:

We know that if the left hand limit and the right hand limit at a given point are different, then the overall limit at that point does not exist. To summarize this idea, since:

[latex]\large{\lim_{x \to 0^-}(\frac{1}{x}) \neq \lim_{x \to 0^+}(\frac{1}{x}) }[/latex]

Then:

[latex]\large{\lim_{x \to 0}(\frac{1}{x})}[/latex] does not exist!

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