Advertisements
Advertisements
Question
Evaluate the following Limits: `lim_(x -> 0)[((5^x - 1)^2)/(x*log(1 + x))]`
Advertisements
Solution
`lim_(x -> 0)((5^x - 1)^2)/(x*log(1 + x))`
= `lim_(x -> 0) ((5^x - 1)^2/x^2)/((x*log(1 + x))/x^2) ...[("As" x -> 0"," x ≠ 0 therefore x^2 ≠ 0),("Divide Numerator and"),("Denominator by " x^2)]`
= `(lim_(x -> 0)((5^x - 1)/x)^2)/(lim_(x -> 0)(log(1 + x))/x`
= `(log 5)^2/1 ...[(lim_(x -> 0) ("a"^x - 1)/x = log "a"","),(lim_(x -> 0) (log(1 + x))/x = 1)]`
= (log 5)2
APPEARS IN
RELATED QUESTIONS
Evaluate the following Limits: `lim_(x -> 0)[(log(4 - x) - log(4 + x))/x]`
Evaluate the following limit :
`lim_(x -> 0) [(6^x + 5^x + 4^x - 3^(x + 1))/sinx]`
Evaluate the following limit :
`lim_(x -> 0) [(8^sinx - 2^tanx)/("e"^(2x) - 1)]`
Evaluate the following limit :
`lim_(x -> 0) [(3 + x)/(3 - x)]^(1/x)`
Evaluate the following limit :
`lim_(x -> 0)[(5x + 3)/(3 - 2x)]^(2/x)`
Evaluate the following limit :
`lim_(x ->0) [("a"^x - "b"^x)/(sin(4x) - sin(2x))]`
The value of `lim_{x→0}{(a^x + b^x + c^x + d^x)/4}^{1/x}` is ______
If f: R → R is defined by f(x) = [x - 2] + |x - 5| for x ∈ R, then `lim_{x→2^-} f(x)` is equal to ______
`lim_(x -> 0) (sin^4 3x)/x^4` = ________.
The value of `lim_{x→0} (1 + sinx - cosx + log_e(1 - x))/x^3` is ______
Evaluate the following:
`lim_(x->0)[((25)^x -2(5)^x+1)/x^2]`
Evaluate the following `lim_(x->0)[((25)^x - 2(5)^x+1) /(x^2)]`
Evaluate the following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the following limit :
`lim(x>2)[(z^2 -5z+6)/(z^2-4)]`
Evaluate the following limit :
`lim_(x->0)[(sqrt(6+x+x^2)-sqrt6)/x]`
Evaluate the following:
`lim_(x->0)[((25)^x - 2(5)^x + 1)/x^2]`
Evaluate the following:
`lim_(x->0)[((25)^x -2(5)^x + 1)/x^2]`
Evaluate the following:
`lim_(x->0)[((25)^x -2(5)^x +1)/(x^2)]`
\[\lim_{x\to0}\frac{\mathrm{e}^{\tan x}-\mathrm{e}^{x}}{\tan x-x}=\]
