Existence of Limits

In Exerc...

Question

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

limx→0 x/|x|

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Consider the function F of X is equal to -2 x divided by the absolute value of X. Explain why the limit as X approaches 0 of F of X, does not exist. Awesome. So it appears for this particular problem we're asked to take our given function of F of X and explain why the limit as X approaches 0 of F of X does not exist. So we're trying to explain why this limit does not exist. So now that we know what we're ultimately trying to solve for, First off, we need to determine once again whether or not the limit of F of X is equal to -2 X divided by the absolute value of X as X approaches 0 exists. And we need to, in order to do this, we need to evaluate the left hand and right hand limits separately. So let us start with the right hand limit, and this is as X approaches 0 from the right. That's what the plus sign and the power means. So as X approaches 0 from the right, So this is when X is greater than 0, the absolute value function will simplify to the absolute value of X will be equal to X simply. Fantastic. So substituting these values back into our equation for F of X, we will find that F of X is equal to -2 X divided by X which is equal to -2. So, therefore, the limit as X approaches 0 from the right of F of X is going to be equal to -2 simply. Now, let's evaluate the left hand limit. So this is as X approaches 0 from the left. That's what that negative symbol in the power means. And this is where X is less than 0, and the absolute value function will will simplify to the absolute value of X. It's gonna be equal to negative X. So substituting these pieces of information back into our equation for F of X, we will find that F of X is equal to negative. 2 X divided by negative X, which will be simplify and solved to simply 2. So therefore the limit as X approaches 0 from the left of F of X is going to be equal to 2. So, therefore, without a doubt, since the limit as X approaches 0 from the right. Of F of X cannot equal the limit as X approaches 0 from the left of F of X. That will mean that the limit As X approaches 0 of F of X does not exist. And that is our final function. So our final, so that's our final expression for our final answer using our function. So we know we have now explained why the limit does not exist. So to quickly summarize, our final answer without a doubt has to be simply The limit as X approaches 0 of F of X does not exist because the limit as X approaches 0 from the right of F of X cannot equal the limit as X approaches 0 from the left of F of X. And that's it. That is our final answer. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Existence of Limits

In Exercises 5 and 6, explain why the limits do not exist.

limx→0 x/|x|

Key Concepts

Limits

One-Sided Limits

Absolute Value Function

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