What Limits Are Undefined in Calculus?

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In the world of calculus, certain limits are going to be considered undefined. Learn about what limits are undefined in calculus with help from an experienced math tutor in this free video clip.

Part of the Video Series: Calculus Explained
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Video Transcript

Hi there. This is Ryan Malloy here at the Worldwide Center of Mathematics. In this video, we're going to discuss what limits are undefined in calculus. So here we have two example functions. In green we have f of x is equal to one over x. And in the purple, we have g of x is equal to the cosine of x. One of the trigonometric functions. We'll take a look at the green one first. Suppose that we're asked the limit as x approaches zero of f of x. Well a slight problem here is that the function approaches different values as you approach zero from different directions. For example, if you take the one sided limit from the left, you get negative infinity. But when you take the one side limit from the right, you get positive infinity. And any time the left and right hand limits don't agree, then the two-sided limit does not exist. Or you could say it's undefined. The second example, we have our trigonometric function cosine. Now we can take the limit at any point in the domain, but what happens if we try to compute the limit as x goes to infinity of g of x. Well this is typically not a problem for most functions whose behavior is well defined. Either shoots up towards positive infinity, shoots down towards negative infinity, or approaches some value asymptotically. But g of x doesn't do any of those things. Cosine just oscillates between zero and one out until infinity. Never approaches one value. So this does not exist. So again, two examples of functions with limits that don't exist. If there is a discontinuity at which the function approaches two different values from different directions or if the end behavior oscillates, does not approach a single value, then those limits will be undefined. My name is Ryan Malloy, and we've just discussed what limits are undefined in calculus.


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