If ‘f' is an even function, prove that `lim_(x rarr 0^-) f(x)= lim_(x rarr 0^+) f(x)` .

Prove that : lim_(x rarr 0^+)(x)/(|x|)=1 .

lim_(x rarr 0) (|x|)/x is :

Prove that : lim_(x rarr 0^-)(x)/(|x|)= -1 .

lim_(x rarr 0)(tan x)/(x) is :

Prove that lim_(x rarr 0) ((1+x)^n-1)/(x)=n .

lim_(x rarr 0)(sin ax)/(bx) is :

lim_(x rarr 0) ((x+1)^5-1)/x is :

Let f(x) is a function continuous for all x in R except at x = 0 such that f'(x) lt 0, AA x in (-oo, 0) and f'(x) gt 0, AA x in (0, oo) . If lim_(x rarr 0^(+)) f(x) = 3, lim_(x rarr 0^(-)) f(x) = 4 and f(0) = 5 , then the image of the point (0, 1) about the line, y.lim_(x rarr 0) f(cos^(3) x - cos^(2) x) = x. lim_(x rarr 0) f(sin^(2) x - sin^(3) x) , is a. ((12)/(25),(-9)/(25)) b. ((12)/(25),(9)/(25)) c. ((16)/(25),(-8)/(25)) d. ((24)/(25),(-7)/(25))

Prove that lim_(x rarr 0) (log(1+x^3))/(sin^3 x)=1 .

If f(x)={(mx^2+n, x 1):} . For what integers m and n does both lim_(x rarr 0)f(x) and lim_(x rarr 1)f(x) exist?