This is really a question about limits, more than about calculus. In his proof that differentiability implies continuity, the professor seems to move the expression f(x0) and then later the expression (x-x0)/(x-x0) from outside the the limit expression to inside it? Does this have to do with f(x0) being a constant?

Question

This is really a question about limits, more than about calculus. In his proof that differentiability implies continuity, the professor seems to move the expression f(x0) and then later the expression (x-x0)/(x-x0) from outside the the limit expression to inside it?  Does this have to do with f(x0) being a constant?

anonymous · Answer

The expression f(x0) is actually a number, so, yes, it can go in and out of a limit. Multiplying and dividing by the same "number" doesn't change anything. In the latter case, in the (x-x0)/(x-x0) expression, the two cancel each other, so nothing is actually changed in the original expression. Hope this answers your question.