If `f(x)={sinx ,x!=npi,n in I2,ot h e r w i s e` `g(x)={x^2+1,x!=0,4,x=0 5,x=2` then `("lim")_(xvec0)g{f(x)}i s=`

If f(x)={(sinx, x != npi " and " n in I_2), (2, " x=npi):} and g(x)={(x^2+1, x != 0),(4,x=0), (5, x=2):} then lim_(x->0) g{f(x)} is

f(x)={x ,xlt=0 1,x=0,t h e nfin d("lim")_(xvec0)f(x)x^2,x >0ife xi s t s

Let f(x)={x+1,x >0, 2-x ,xlt=0 and g(x)={x+3,x 0)g(f(x)).

If f(x)=1/x ,g(x)=1/(x^2), and h(x)=x^2 , then (A) f(g(x))=x^2,x!=0,h(g(x))=1/(x^2) (B) h(g(x))=1/(x^2),x!=0,fog(x)=x^2 (C) fog(x)=x^2,x!=0,h(g(x))=(g(x))^2,x!=0 (D) none of these

If f(x)=sinx, g(x)=x^2 and h(x)=log x, find h[g{f(x)}].

If f(x)={(x^nsin(1/(x^2)),x!=0), (0,x=0):} , (n in I) , then (a) lim_(xrarr0)f(x) exists for n >1 (b) lim_(xrarr0)f(x) exists for n<0 (c) lim_(xrarr0)f(x) does not exist for any value of n (d) lim_(xrarr0)f(x) cannot be determined

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

Let f(x) be a function defined on (-a ,a) with a > 0. Assume that f(x) is continuous at x=0a n d(lim)_(xvec0)(f(x)-f(k x))/x=alpha,w h e r ek in (0,1) then a. f^(prime)(0^+)=0 b. f^(prime)(0^-)=alpha/(1-k) c. f(x) is differentiable at x=0 d. f(x) is non-differentiable at x=0

If f(x)={x-1,xgeq 1 2x^2-2,x 0-x^2+1,xlt=0,a n dh(x) =|x|,t h e n lim_(x->0)f(g(h(x))) is___

If f(x)=|x-2| a n d g(x)=f[f(x)],t h e n g^(prime)(x)= ______ for x>20