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प्रश्न
Evaluate the following limit.
`lim_(x -> 0) ((x+1)^5 - 1)/x`
उत्तर
`lim_(x → 0)((x + 1)^5 - 1)/x`
= `lim_(x → 0)((1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5) - 1)/x`
= `lim_(x → 0) (x(5 + 10x + 10x^2 + 5x^3 + x^4))/x`
= `lim_(x → 0)(5 + 10x + 10x^2 + 5x^3 + x^4)`
= 5
Alternative method: We know:
`lim_(x → a)(x^n - 1)/(x - 1) = na^(n - 1)`
`lim_(x → 0) ((x + 1)^5 - 1)/x`
= `lim_(x → 0) ((x + 1)^5 - 1)/((x + 1) - 1)`
= `lim_(x → 0) 5(x + 1)^4`
= 5
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