Advertisements
Advertisements
Question
Evaluate the following limit.
`lim_(x -> 0) ((x+1)^5 - 1)/x`
Solution
`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
APPEARS IN
RELATED QUESTIONS
Evaluate the following limit.
`lim_(x -> 3) x + 3`
Evaluate the following limit.
`lim_(x -> pi) (x - 22/7)`
Evaluate the following limit:
`lim_(r -> l) pir^2`
Evaluate the following limit.
`lim_(x -> 4) (4x + 3)/(x - 2)`
Evaluate the following limit.
`lim_(x-> -1) (x^10 + x^5 +1)/(x -1)`
Evaluate the following limit.
`lim_(x- > 2) (3x^2 - x - 10)/(x^2- 4)`
Evaluate the following limit.
`lim_(x -> 3) (x^4 - 81)/(2x^2-5x - 3)`
Evaluate the following limit.
`lim_(x-> 0) (ax + b)/(cx + 1)`
Evaluate the following limit.
`lim_(z -> 1) (z^(1/3) - 1)/(z^(1/6) -1)`
Evaluate the following limit.
`lim_(x -> 1) (ax^2 + bx + c)/(cx^2 + bx + a), a+b+c != 0`
Evaluate the following limit.
`lim_(x -> -2) (1/x + 1/2)/(x + 2)`
Evaluate the following limit.
`lim_(x -> 0) (sin ax)/ (bx)`
Evaluate the following limit.
`lim_(x -> 0) (sin ax)/(sin bx), a, b != 0`
