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

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

Evaluate lim_(xto0^(+)) x^(x)"and "lim_(xto0^(+))x^(x^(x))

Prove that Lim_(x to 0) (|x|)/x , x != 0 does not exist .

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(x) = (x^(2)-9)/(x-3) , find it Lim_(x to 3) f(x) exists .

Let f : R to R be a real function. The function f is double differentiable. If there exists ninN and p in R such that lim_(x to oo)x^(n)f(x)=p and there exists lim_(x to oo)x^(n+1)f(x) , then lim_(x to oo)x^(n+1)f'(x) is equal 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) 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 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.