Jika f(x) = 1/2x^2 maka lim x mendekati 0 (f(x+t) - f(x))/t adalah

Jika f(x) = 1/2x^2 maka lim x mendekati 0 (f(x+t) – f(x))/t adalah

Jawaban:

Penjelasan:

f(x + t) = $frac{1}{2(x+t)^{2} } = frac{1}{2(x^{2} + 2xt + t^{2}) }$

f(x) = $frac{1}{2x^{2} }$

f(x + t) – f(x) = $frac{1}{2(x^{2} + 2xt + t^{2}) } - frac{1}{2x^{2} } = frac{x^{2} - (x^{2} + 2xt + t^{2})}{2x^{2} (x^{2} +2xt+t^{2}) } = frac{x^{2} - x^{2} - 2xt - t^{2}}{2x^{2} (x^{2} +2xt+t^{2}) } = frac{ - 2xt - t^{2}}{2x^{4} +4x^3t+2x^2t^{2} }$

$frac{f(x+t) - f(x)}{t} = frac{ - 2xt - t^{2}}{2x^{4} +4x^3t+2x^2t^{2} }.{frac{1}{t} }$

$frac{f(x+t) - f(x)}{t} = frac{ - 2x - t}{2x^{4} +4x^3t+2x^2t^{2} }$

$lim_{t to 0} frac{ - 2x - t}{2x^{4} +4x^3t+2x^2t^{2} } = frac{ - 2x - 0}{2x^{4} +4x^3(0)+2x^2(0^{2}) } = frac{- 2x}{2x^{4} +0+0 } = frac{- 2x}{2x^{4} }$

$lim_{t to 0} frac{ - 2x - t}{2x^{4} +4x^3t+2x^2t^{2} } = frac{- 1}{x^{3} }$ (B)