Samudro’s Problem

Diket:

 f (x)=frac{(1-x)(4-x)}{x}

Tentukan nilai dari

lim_{hto0}  frac{f (x+h) - f (x)}{h}


Solution:

f (x+h) = frac{[1-(x+h)][4-(x+h)]}{x}=frac{4- (x+h)-4 (x+h)+(x+h)^2}{x}

f (x+h)=frac{4- x-h-4x-4h+x^2 + 2xh + h^2}{x}=frac{4-5x-5h+x^2+2xh+h^2}{x}

 

Now,

 

f (x+h)-f (x)=frac{4-5x-5h+x^2+2xh+h^2 - (1-x)(4-x)}{x}

f (x+h)-f (x)=frac{4-5x-5h+x^2+2xh+h^2 - (4 -x-4x+x^2)}{x}

f (x+h)-f (x)=frac{4-5x-5h+x^2+2xh+h^2 - 4 +5x-x^2}{x}

f (x+h)-f (x)=frac{-5h+2xh+h^2}{x}

 

The expression inside the limit will be:

frac {f (x+h)-f (x)}{h}=frac{-5h+2xh+h^2}{hx}=frac {-5+2x+h}{x}

Finally,

 

lim_{hto0} frac {-5 +2x+h}{x} = frac {-5+2x}{x}