Evaluate: `("lim")_(nvec0)(2^x-1-x)/(x^2),` without using LHospitals rule and expansion f the series.

Evaluate ("Lim")_(x->0)(2^x-1)/((1+x)^(1//2)-1) using Lhospitals Rule

Evaluate : lim_(x rarr0)(a^(2x)-1)/(x)

Evaluate ("Lim")_(x->0)[1/(x^2)-1/(sin^2x)] using LHospitals Rule or otherwise :

Evaluate ("Lim")_(x->0)(1+sinx-cosx+ln(1-x))/(x tan^2x) using LHospitals Rule or otherwise :

Evaluate ("Lim")_(x->pi/4)(2-tanx)^(1/(ln(tanx))) using Lhospitals Rule

Evaluate lim (x-sin x)/(x^(3)). (Do not use either LHospitals rule or series expansion for sin x. . Hence,evaluate lim_(n rarr0)(sin x-x cos x+x^(2)cot x)/(x^(5))

(i) Let h (x) = underset(x to oo)lim(x^(2n) f(x) + g(x))/(1+x^(2n)) , find h(x) in terms of f(fx) and g(x) (ii) without using L Hospital rule or series expansion for e^(x) evaluate underset(x to 0) lim (e^(x) -1-x)/x^(2) (iii) underset(n to oo) lim [ e^(1/n)/n^(2) + 2 ((e^(1/n))^(2))/n^(2) + 3. ((e^(1/n))^(3))/n^(2)+.......+ n((e^(1/n))^(n))/n^(2)] (iv) underset(x to 0)lim[ (a sin x)/x ] + [ (b tan x)/x] Where a,b are inegers and [] denotes integral part. (v) underset(x to a)lim (sinx/sina)^(1/(x-a))

Evaluate ("Lim")_(x->0)(logcosx)/x using Lhospitals Rule

Evaluate ("Lim")_(x->pi/2)tanxlogsinx using Lhospitals Rule

Prove that lim_(x rarr0)(f(x+h)+f(x-h)-2f(x))/(h^(2))=f^(x) (without using Ll'Hospital srule).