limit as x approaches a of (x^5-a^5)/(x^4-a^4)

解

x \to a lim (\frac{x ^{5} - a ^{5}}{x ^{4} - a ^{4}})

\frac{5 a}{4}

解答ステップ

x \to a lim (\frac{x ^{5} - a ^{5}}{x ^{4} - a ^{4}})

簡素化

\frac{x ^{5} - a ^{5}}{x ^{4} - a ^{4}} : \frac{x ^{4} + a x ^{3} + a ^{2} x ^{2} + a ^{3} x + a ^{4}}{( x ^{2} + a ^{2} ) ( x + a )}

= x \to a lim (\frac{x ^{4} + a x ^{3} + a ^{2} x ^{2} + a ^{3} x + a ^{4}}{( x ^{2} + a ^{2} ) ( x + a )})

値を挿入する

x = a

= \frac{a ^{4} + a a ^{3} + a ^{2} a ^{2} + a ^{3} a + a ^{4}}{( a ^{2} + a ^{2} ) ( a + a )}

簡素化

\frac{a ^{4} + a a ^{3} + a ^{2} a ^{2} + a ^{3} a + a ^{4}}{( a ^{2} + a ^{2} ) ( a + a )} : \frac{5 a}{4}

= \frac{5 a}{4}