### Video Transcript

If a function π is differentiable at π₯ is equal to two, which of the following must be true? I) The limit as π₯ approaches two of π of π₯ exists. II) π of two exists. And III, π prime of two exists.

Weβre told that our function π is differentiable at π₯ is equal two. And there are a couple of things we know about the conditions that need to be satisfied for a function to be differentiable at a point. Weβll look at these conditions for our function and compare with the given options I, II, and III. The first thing we can say about a differentiable function is π of π₯ is differential at the point π₯ equal to π, if π prime of π exists.

Weβre told in the question that our function π is differentiable at π₯ is equal to two. If we let π equal to two in our condition for differentiable function, then π prime of π, which is π prime of two, exists. So by satisfying this condition of a differentiability, we can see that option III is true. π prime of two exists.

The second thing we know about functions which are differentiable at a point is that if π of π₯ is differentiable at π₯ equal to π, then π of π₯ is continuous at π₯ is equal to π. And to be continuous at a point, π of π₯ must satisfy three conditions. One, that π exists at the point π₯ equal to π. Two, that the limit as π₯ approaches π of π of π₯ exists. And three, that the limit as π₯ approaches π of π of π₯ is equal to π of π.

Now, we know that our function π is differentiable at π₯ is equal to two. So it must be continuous at π₯ is equal to two. So if we let π equal two, then by our first condition for continuity, π of π exists, so π of two exists. This means that our option II is also true.

If we again let π equal to two in our second condition for continuity, we know that our function is differentiable at π₯ is equal to two and therefore must be continuous at π₯ is equal to two. So the second condition for continuity tells us that the limit as π₯ approaches two of π of π₯ exists. And this means that our option I is also satisfied. Using the conditions for differentiability and continuity at a point, we can therefore say that options I, II, and III are all true. This means that the limit as π₯ approaches two of π of π₯ exists, π of two exists, and π prime at π₯ is equal to two also exists.

We can also note that since a function is differentiable at π₯ is equal to two, the limit as π₯ approaches two of π of π₯ is equal to π of two, which satisfies condition three for continuity. Itβs also worth noting that our second point of π of π₯ is differentiable at π₯ is equal to π. Then π of π₯ is continuous at π₯ is equal to π is not necessarily true in the opposite direction. This means that if a function is continuous at a point, that doesnβt necessarily mean that itβs differentiable at that point.