Question

If \[f\] is differentiable at \[c\], which limit equals \[f'(c)\]?

Options

• \[\lim_{x\to c}[f(x)-f(c)]=f'(c)\]

• \[\lim_{x\to c}\frac{f(x)+f(c)}{x-c}=f'(c)\]

• \[\lim_{x\to c}\frac{f(x)-f(c)}{x-c}=f'(c)\]

• \[\lim_{x\to c}\frac{f(x)-f(c)}{x+c}=f'(c)\]

Answer 1

Differentiability at \[c\] gives the difference-quotient limit with denominator \[x-c\]. That limit is precisely the derivative \[f'(c)\].