`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`

If f(x)={{:(x","" "xlt0),(1","" "x=0),(x^(2)","" "xgt0):}," then find " lim_(xto0) f(x)" if exists.

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-|x|)/(x)","xne0),(2", "x=0):}, show that lim_(xto0) f(x) does not exist.

If f(x)=cosx-int_0^x(x-t)f(t)dt ,t h e nf^(primeprime)(x)+f(x) is equal to (a) -cosx (b) -sinx (c) int_0^x(x-t)f(t)dt (d) 0

f(x) is polynomial function of degree 6, which satisfies ("lim")_(x_vec_0)(1+(f(x))/(x^3))^(1/x)=e^2 and has local maximum at x=1 and local minimum at x=0a n dx=2. then 5f(3) is equal to

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

If lim_(xto0) [1+x+(f(x))/(x)]^(1//x)=e^(3) , then the value of ln(lim_(xto0) [1+(f(x))/(x)]^(1//x)) is _________.