`lim_(x-gt0)((1-x)^n-1)/x`

lim_(x-gt0) (e^(3x)-1)/(log(1+5x))

Evaluate the limits, if exist lim_(x-gt0)(log_e(1+2x))/(x)

(i) lim_(x to 0) (a^(x) - 1)/(log_(a)(1 + x)), a gt 0 (ii) lim__(x to 0) (In (X + a)- In a)/(e^(2x) - 1) (ii) lim_(x to (pi)/(4)) (In(tanx))/(1 - cotx)

(i) lim_(x to 0) (a^(x) - 1)/(log_(a)(1 + x)), a gt 0 (ii) lim__(x to 0) (In (X + a)- In a)/(e^(2x) - 1) (ii) lim_(x to (pi)/(4)) (In(tanx))/(1 - cotx)

lim_(x rarr0)((x+1)^(n)-1)/(x)

lim_(x rarr0)((x+1)^(n)-1)/(x)

Statement-1 : lim_(x to 0) (1 - cos x)/(x(2^(x) - 1)) = (1)/(2) log_(2) e . Statement : lim_(x to 0) ("sin" x)/(x) = 1 , lim_(x to 0) (a^(x) - 1)/(x) = log a, a gt 0

Statement-1 : lim_(x to 0) (1 - cos x)/(x(2^(x) - 1)) = (1)/(2) log_(2) e . Statement : lim_(x to 0) ("sin" x)/(x) = 1 , lim_(x to 0) (a^(x) - 1)/(x) = log a, a gt 0

If f(x)=lim_(n rarr oo)(1+x)^(n) comment on the continuity of f(x)atx=0 and explain lim_(x rarr0)(1+x)^((1)/(x))=e