Question

Find lim as h→0 f(x+h)-f(x)/h

a. f(x) = 1-x/x+2

b. f(x) = sqrt x²-3

Answer 1

(a) $\lim\nolimits_h\to0\lbrack\frac{f(x+h)-f(x)}{h}\rbrack$

$=\lim\nolimits_h\to0\lbrack\frac{{\frac{1-(x+h)}{x+h+2}}-{\frac{1-x}{x+2}}}{h}\rbrack$

$=\lbrack\frac{{\frac{1-(x+h)}{x+h+2}}-{\frac{1-x}{x+2}}}{h}\rbrack$

$=\lim\nolimits_h\to0\lbrack\frac{((x+2)(1-x-h))-((1-x)(x+h+2))}{h(x+h+2)(x+2)}\rbrack$

$=\frac{3}{2\sqrt{-x+1}\left((x+2)^{\smash{\frac{3}{2}}}\right)}$

(b) the derivative of f(x) with respect to x is defined as:

$\lim\nolimits_h\to0\lbrack\frac{f(x+h)-f(x)}{h}\rbrack$

= $\lim\nolimits_h\to0\lbrack\frac{\sqrt{\left((x+h)^{\smash{2}}\right)-3)}-\sqrt{\left((x)^{\smash{2}}-3\right)}}{h}\rbrack$

$=\lbrack\frac{\sqrt{\left((x+h)^{\smash{2}}\right)-3)}-\sqrt{\left((x)^{\smash{2}}-3\right)}}{h}\rbrack$

$=\lim\nolimits_h\to0\lbrack\frac{\sqrt{\left((x+h)^{\smash{2}}\right)-3)}-\sqrt{\left((x)^{\smash{2}}-3\right)}}{h}\rbrack\lbrack\frac{\sqrt{\left((x+h)^{\smash{2}}\right)-3)}+\sqrt{\left((x)^{\smash{2}}-3\right)}}{\sqrt{\left((x+h)^{\smash{2}}\right)-3)+\sqrt{\left((x^{\smash{2}}\right)-3)}}}\rbrack$

$=\frac{x}{\sqrt{\left(x^{\smash{2}}\right)-3}}$