Question Video: Differentiability of a Function Mathematics

Video Transcript

Consider the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥. Find the limit as ℎ approaches zero from the right of 𝑓 of ℎ divided by ℎ. Find the limit as ℎ approaches zero from the left of 𝑓 of ℎ divided by ℎ. What can you conclude about the derivative of 𝑓 of 𝑥 at 𝑥 is equal to zero?

The question gives us the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥. The first thing it wants us to do is evaluate this limit. That’s the limit as ℎ approaches zero from the right of 𝑓 of ℎ divided by ℎ. And of course, we know that 𝑓 of ℎ is equal to the absolute value of ℎ. Now, since our limit is as ℎ is approaching zero from the right, our values of ℎ will be strictly greater than zero. However, if our values of ℎ are greater than zero, our values of ℎ are positive. And this means the absolute value of ℎ is just equal to ℎ. So this gives us the limit as ℎ approaches zero from the right of ℎ divided by ℎ.

We then cancel the shared factor of ℎ in our numerator and our denominator. We can do this because this gives us a new function which is exactly equal to ℎ divided by ℎ everywhere except where ℎ is equal to zero. This means their limits as ℎ approaches zero from the right will be the same. This gives us the limit as ℎ approaches zero from the right of the constant one, and we know that this is just equal to one. So, we’ve evaluated our first limit; it’s equal to one.

We can do something similar to evaluate our second limit, the limit as ℎ approaches zero from the left of 𝑓 of ℎ divided by ℎ. Again, 𝑓 of ℎ is the absolute value of ℎ. And since our limit is as ℎ is approaching zero from the left, our values of ℎ will be strictly less than zero. This time, since our values of ℎ are less than zero, our values of ℎ are negative. This tells us the absolute value of ℎ would just be equal to negative ℎ. So this gives us the limit as ℎ approaches zero from the left of negative ℎ divided by ℎ.

Again, we’ll cancel the shared factor of ℎ in our numerator and our denominator. And this gives us the limit as ℎ approaches zero from the left of negative one, and negative one is a constant. So, this limit just evaluates to give us negative one. So, we’ve now evaluated our second limit; it’s equal to negative one.

The final part of this question wants us to make a concluding statement about the derivative of our function 𝑓 of 𝑥 when 𝑥 is equal to zero. Let’s start by recalling the definition of the derivative of a function at a point. We recall the derivative of some function 𝑔 of 𝑥 at the point 𝑥 zero is defined as the limit as ℎ approaches zero of 𝑔 evaluated at 𝑥 zero plus ℎ minus 𝑔 evaluated at 𝑥 zero divided by ℎ if this limit exists.

We want to find the derivative of our function 𝑓 of 𝑥 is equal to the absolute value of 𝑥 when 𝑥 is equal to zero. So, we’ll set 𝑔 of 𝑥 to be the absolute value of 𝑥 and 𝑥 zero to be equal to zero. Substituting these into our definition of the derivative, we get that the derivative of the absolute value of 𝑥 when 𝑥 is equal to zero is defined to be. The limit as ℎ approaches zero of the absolute value of zero plus ℎ minus the absolute value of zero divided by ℎ if this limit exists. And we can simplify this limit. First, zero plus ℎ is just equal to ℎ. Next, the absolute value of zero is just equal to zero.

So, the derivative of the absolute value of 𝑥 when 𝑥 is equal to zero is equal to the limit as ℎ approaches zero of the absolute value of ℎ divided by ℎ if this limit exists. But remember what we showed in the first two parts of this question. In the first part, we showed the limit as ℎ approaches zero from the right of the absolute value of ℎ divided by ℎ is equal to one. However, in our second part, we showed the limit as ℎ approaches zero from the left of the absolute value of ℎ divided by ℎ was equal to negative one. The left and right limits were not equal. And this tells us that this limit does not exist.

And if this limit does not exist, then that means that the derivative of 𝑓 of 𝑥 when 𝑥 is equal to zero also doesn’t exist. Therefore, for the function 𝑓 of 𝑥 is equal to the absolute value of 𝑥, we can conclude that the derivative of 𝑓 of 𝑥 at 𝑥 is equal to zero does not exist because the left-hand and right-hand limits of its derivative at this point are unequal.