Question

Multiple Choice question: why is the answer B? I would think it's C.

Here's the question.

Why is the answer B? I know that the derivative cannot exist at 3, but I would think that the left limit is still 2, since even though there is a hole on the left function, the limit is about the value that is being approached. So the slope to the left of 3 would still be 2, no? Please explain why it is not C.

Thank you so much!

Answer 1

Please see below.

Explanation:

As `hrarr0^-`, the value of `3+h` is less that 3.

So, the secant joining `(x+h,f(x+h))` to the point `(3,f(3)) = (3,4)` is joining a point on `y=2x-2` to a point on `2x-4`.

As `hrarr0^-`, the secant line becomes vertical and the slope decreases without bound.

The graph below shows `f(x)` in blue and three secant lines. You can see that the secant line through the point closest to `x=3` (not labelled, but it's the red line) has very negative slope. It is nearly vertical.