Let `f(x)={x+1,\ if\ x >0, and x-1,\ if\ x<0dot` Prove that `("lim")_(x->1)\ f(x)` does not exist.

Let f(x)=\ {x+5,\ if\ x >0 and x-4,\ if\ x 0)\ f(x) does not exist.

{:("Column-I","Column-II"),(A.f(x) = (1)/(sqrt(x -2)),p.lim_(x to 0)f(x) =1),(B. f(x) = (3x - "sin"x)/(x + "sin" x), q. lim_(x to 0)f(x) = 0),(C.f(x) = x "sin"(pi)/(x) f(0)=0,r.lim_(x to oo) f(x) = 0),(f(x) = tan^(-1) (1)/(x),s.lim_(x to 0) "does not exist"):}

Let f (x)= {{:(x ^(n) (sin ""(1)/(x )",") , x ne 0),( 0"," , x =0):} Such that f (x) is continuous at x =0, f '(0) is real and finite, and lim _(x to 0^(+)) f'(x) does not exist. The holds true for which of the following values of n ?

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

Letf(x)={{:(x+1,", "if xge0),(x-1,", "if xlt0):}".Then prove that" lim_(xto0) f(x) does not exist.

Consider the following graph of the function y=f(x). Which of the following is//are correct? (a) lim_(xto1) f(x) does not exist. (b) lim_(xto2)f(x) does not exist. (c) lim_(xto3) f(x)=3. (d)lim_(xto1.99) f(x) exists.

Let f(x) = (x)/(1+|x|), x in R , then f is

If f(x)={:{(1+sin x", " 0 le x lt pi//2),(" "1 ", " x lt0):} then show that f'(0) does not exist.

Let f(x) be a function defined by f(x) = {{:((3x)/(|x|+2x),x ne 0),(0,x = 0):} Show that lim_(x rarr 0) f(x) does not exist.

Let f(x)={[x]x in I x-1x in I (where [.] denotes the greatest integer function) and g(x)={sinx+cosx ,x<0 1,xgeq0 . Then for f(g(x))a tx=0 (lim)_(xvec0)f(g(x)) exists but not continuous Continuous but not differentiable at x=0 Differentiable at x=0 (lim)_(xvec0)f(g(x)) does not exist f(x) is continuous but not differentiable