I don't get how the concept of limits allows us to move from secants to tangents. From what I've gathered, to find the derivative/tangent line at a point, we take a secant line, move the points infinitely close to each other, and find the slope value that it is getting infinitely close to. That slope value will be the limit, the derivative, the slope of the tangent line.
What I don't get is how we can make this logical leap, that because the limit as x approaches a is approaching a certain value, the value at a (the derivative) must be the value of a limit.
Lim x->4=0. But at x=4, the value won't be 0! So how can we make the assumption that because the secant shrinks and gets closer and closer to being a tangent, lim x->a approaches a value, then at a, the derivative will be that value?
I have a gut feeling that the reason why it works out is because all differentiable functions are continuous. But can you elaborate and explain on this statement? Because I don't see how I can justify my intuition. Can someone someone prove/show to me that because all differentiable functions are continuous, when lim x->a approaches a value, when actually at that value, the derivative will be equals to that?
But then if continuity is the thing that makes it work, how come not all continuous functions are differentiable?
Can someone please explain the above to me mathematically? Is there a way to explain this, without epsilon delta proofs, to someone learning Khan Academy Calculus?