If `f(x)=(cosx)/((1-sinx)^(1/3))` then (a) `("lim")_(xrarrpi/2)f(x)=-oo` (b) `("lim")_(xrarrpi/2)f(x)=oo` (c) `("lim")_(xrarrpi/2)f(x)=o (d) none of these

Let f(x)=(x^2-9x+20)/(x-[x]) (where [x] is the greatest integer not greater than xdot Then (a) ("lim")_(xvec5)f(x)=1 (b) ("lim")_(xvec5)f(x)=0 (c) ("lim")_(xvec5)f(x)does not exist (d)none of these

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

If f(x)=lim_(nrarroo) (cos(x)/(sqrtn))^(n) , then the value of lim_(xrarr0) (f(x)-1)/(x) is

If ("lim")_(xtoa)[f(x)g(x)] exists, then both ("lim")_(xtoa)f(x)a n d("lim")_(xtoa)g(x) exist.

Given ("lim")_(xvec0)(f(x))/(x^2)=2,w h e r e[dot] denotes the greatest integer function, then (a) ("lim")_(xvec0)[f(x)]=0 (b) ("lim")_(xvec0)[f(x)]=1 (c) ("lim")_(xvec0)[(f(x))/x] does not exist (d) ("lim")_(xvec0)[(f(x))/x] exists

Let f(x)=|x-1|dot Then (a) f(x^2)=(f(x))^2 (b) f(x+y)=f(x)+f(y) (c) f(|x|)-|f(x)| (d) none of these

If ("lim")_(xveca)[f(x)+g(x)]=2a n d ("lim")_(xveca)[f(x)-g(x)]=1, then find the value of ("lim")_(xveca)f(x)g(x)dot

Which of the following true ({.} denotes the fractional part of the function)? (a) ("lim")_(xtooo)((log)_e x)/({x})=oo (b) ("lim")_(xto2^+)x/(x^2-x-2)=oo (c) ("lim")_(xto1^-)x/(x^2-x-2)=-oo (d) ("lim")_(xtooo)((log)_(0 .5) x)/({x})=oo

Sketch the graph of an example of a rational function f that satisfies all the given conditions. (i) f(0)=0,f(1), lim_(xrarroo) f(x)=0,f " is odd" (ii) lim_(xrarr0^(+)) f(x)=oo,lim_(xrarr0^(-)) f(x)=-oo,lim_(xrarroo) f(x)=1,lim_(xrarr-oo) f(x)=1 (iii) lim_(xrarr-2) f(x)=oo, lim_(xrarr-oo) f(x)=3,lim_(xrarroo) f(x) = 3 , f(0)=0

If f(x)=(3x^2+a x+a+1)/(x^2+x-2), then which of the following can be correct (a) ("lim")_(xto1)f(x) ex i s t s a=-2 (b) ("lim")_(xto-2)f(x) ex i s t s a=13 (c) ("lim")_(xto1)f(x)=4/3 (d) ("lim")_(xto-2)f(x)=-1/3