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_(x rarr0^(-))f(x)=lim_(x rarr0^(+))f(x)

If f is an even function, prove that lim_(xto0^(-)) (x)=lim_(xto0^(+)) f(x).

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

If f(x)=|x|, prove that lim_(x rarr0)f(x)=0

Left hand derivative and right hand derivative of a function f(x) at a point x=a are defined as f'(a^-)=lim_(h to 0^(+))(f(a)-f(a-h))/(h) =lim_(hto0^(+))(f(a+h)-f(a))/(h) andf'(a^(+))=lim_(h to 0^+)(f(a+h)-f(a))/(h) =lim_(hto0^(+))(f(a)-f(a+h))/(h) =lim_(hto0^(+))(f(a)-f(x))/(a-x) respectively. Let f be a twice differentiable function. We also know that derivative of a even function is odd function and derivative of an odd function is even function. The statement lim_(hto0)(f(-x)-f(-x-h))/(h)=lim_(hto0)(f(x)-f(x-h))/(-h) implies that for all x"inR ,

If f is an even function such that lim_(h rarr0)(f(h)-f(0))/(h) has some finite non-zero value,then prove that f(x) is not differentiable at x=0.

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.

Given lim_(x rarr0)(f(x))/(x^(2))=2, where [.] denotes the greatest integer function,then lim_(x rarr0)[f(x)]=0lim_(x rarr0)[f(x)]=1lim_(x rarr0)[(f(x))/(x)] does not exist lim _(x rarr0)[(f(x))/(x)] exists

Let f:R rarr R be a function satisfying Lim_(xrarr0^(+))f(x)=2 , Lim_(xrarr0^(-)) f(x)=3 and f(0)=4 The value of (Lim_(xrarr0) f(x^(3)-x^(2)))/(Lim_(xrarr0) f(2x^(4)-x^(5)))=