`f(x)=xsin(1/x),x ne0=0,x=0` examine whether f'(0)exists.

A function f(x) is defined as follows : " "f(x)={:{(x-1",","when "x gt 0),(-(1)/(2)",","when "x = 0),( x+1",","when "xlt0):} Draw the graph of the function f(x). From the graph find the value of f(x) at x=-(1)/(2) and examine whether f(x) is continuous at x = 0.

Let f(x)={(x exp[-(1/|x|+1/x)], x ne 0),(0, x=0):} Test whether f(x) is differentiable at x=0

If f(x)={{:(x+sinx,"when "xlt0),(0,"when "xge0):} examine whether f(x) is continuous at x = 0.

Draw the graph of y=f(x)=(x^(2))/(x) and from the graph examine whether underset(xrarr0)"lim"f(x) exists or not .

Let , f(x)=1/xsin(x^2) when x ne0 =0 when x=0 Discuss the continuity and differentiability of f(x) at x=0.

Let f(x)={(x^2|cos"pi/x|, x ne0),(0, x=0):} , x in R , then f is

If f(x)={(x"sin"(1)/(x),"when "x ne 0),(0,"when "x =0):} then at x = 0, the function f(x) is -

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

A function f(x) is defined as follows : f(x)=-x," when "x lt0 =2," when "0 lex lt2 =4-x," when "x ge2 Draw the graph of the function and from the graph examine whether f(x) is continuous at x = 0 and x = 2 or not.

If a f(x) +b f((1)/(x))=(1)/(x)-5,x ne 0,a ne b , then overset(2)underset(1)int f(x)dx equals