Prove that `lim_(x->0) (f(x+h)+f(x-h)-2f(x))/h^2=f''(x)` (without using L' Hospital srule).

If f(x),g(x)a n dh(x) are three polynomial of degree 2, then prove that varphi(x)=|f(x)g(x)h(x)f'(x)g'(x h '(x)f' '(x)g' '(x h ' '(x)| is a constant polynomial.

lim_(h->0) (f(2h+2+h^2)-f(2))/(f(h-h^2+1)-f(1)) given that f'(2)=6 and f'(1)=4 does not exist (a) is equal to - 3/2 (b) is equal to 3/2 (c) is equal to 3

If lm_(x->oo) f(x) exists and is finite and nonzero and if lim_(x->oo) {{f(x)+(3f(x)−1)/(f_2(x))}=3 ,then the value of lim_(x->oo) f(x) is

Instead of the usual definition of derivative Df(x), if we define a new kind of derivative D^*F(x) by the formula D*f(x)=lim_(h->0)(f^2(x+h)-f^2(x))/h ,w h e r ef^2(x) mean [f(x)]^2 and if f(x)=xlogx ,then D^*f(x)|_(x=e) has the value (A)e (B) 2e (c) 4e (d) none of these

f(x)={(x-1",",-1 le x le 0),(x^(2)",",0le x le 1):} and g(x)=sinx Consider the functions h_(1)(x)=f(|g(x)|) and h_(2)(x)=|f(g(x))|. Which of the following is not true about h_(2)(x) ?

f(x)={(x-1",",-1 le x le 0),(x^(2)",",0le x le 1):} and g(x)=sinx Consider the functions h_(1)(x)=f(|g(x)|) and h_(2)(x)=|f(g(x))|. Which of the following is not true about h_(1)(x) ?

Given that h(x) =f^(@) g(x) , fill in the table for h (x)

Let f :RtoR be a positive, increasing function with lim_(xtooo) (f(3x))/(f(x))=1 . Then lim_(xtooo) (f(2x))/(f(x)) is equal to

If f is an even function such that (lim)_(hvec0)(f(h)-f(0))/h has some finite non-zero value, then prove that f(x) is not differentiable at x=0.