If `f` is an even function, then prove that `lim_(x->0^-) f(x) = lim_(x->0^+) f(x)`

If f is an even function , then prove that Lim_( xto 0^(-)) f(x) = Lim_(x to 0^(+)) f(x) , whenever they exist .

If |f(x)|lex^(2), then prove that lim_(xto0) (f(x))/(x)=0.

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '(0) equais to :

If f (x) is a thrice differentiable function such that lim _(xto0)(f (4x) -3 f(3x) +3f (2x) -f (x))/(x ^(3))=12 then the vlaue of f '''(0) equais to :

If f(x)=sgn(x)" and "g(x)=x^(3) ,then prove that lim_(xto0) f(x).g(x) exists though lim_(xto0) f(x) does not exist.

If f is an even function such that (lim)_(hvec0)(f(h)-f(0))/h has some finite non-zero value, then prove that f(x) is not differentiable at x=0.

If f is an even function such that (lim)_(hvec0)(f(h)-f(0))/h has some finite non-zero value, then prove that f(x) is not differentiable at x=0.

Given ("lim")_(xvec0)(f(x))/(x^2)=2, \where\ [dot] denotes the greatest integer function, then (A) ("lim")_(xvec0)[f(x)]=0 (B) ("lim")_(xvec0)[f(x)]=1 (C) ("lim")_(xvec0)[(f(x))/x] does not exist (D) ("lim")_(xvec0)[(f(x))/x] exists

Evaluate lim_(xrarr0) f(x) , where

Given a real-valued function f such that f(x)={(tan^2{x})/((x^2-[x]^2) sqrt({x}cot{x}),for x 0 Where [x] is the integral part and {x} is the fractional part of x , then ("lim")_(xvec0^+)f(x)=1 , ("lim")_(xvec0^-)f(x)=cot1 , cot^(-1)(("lim")_(xvec0^-)f(x))^2=1 , tan^(-1)(("lim")_(xvec0^+)f(x))=pi/4