Question

Help with explaining true/false questions? I figured out that they're true of false but need help explaining why. Please help, a medal will be given!

Answer 1

which ones

Answer 2

oh geez I'm only at Pre calc level sorry

Answer 3

1 is false
left hand limit equals right hand limit means the limit exists, but for the function to be continuous there it also has to be defined there as well

Answer 4

a "hole" is a good example of how a function can have the limit exist but not be continuous

Answer 5

So it must fulfill both conditions to be continuous?

Answer 6

yes, it has to exist there as well
i.e.
\[\lim_{x\to a^+}f(x)=\lim_{x\to a^-}f(x)=f(a)\]
the first to can be true even if \(f(a)\) is not the limit

Answer 7

first *two

Answer 8

What if only the first two are true but not f(a)?

Answer 9

then the limit exists but the function is not continuous there
example
\[\lim_{x\to 2}\frac{x^2-4}{x-2}=4\] but for \(f(x)=\frac{x^2-4}{x-2}\) we know \(f(2)\) does not exist

Answer 10

Okay, I that question understand now. Are there any other ones you can help explain?

Answer 11

probably all

Answer 12

2 is true because the derivative is a formula for the slope, so if \(f'(2)=0\) then that means the slope of the tangent line is 0, aka horizontal

Answer 13

3 is true, that is the definition of the derivative of \(f\) at \(a\)