hard question in calculus?

Question

if lim x-->0 f(x)/x^2=1, find lim x-->0 f(x), and lim x--> f(x)/x

Answer 1

[x→0]lim f(x)=0
[x→0]lim f(x)/x=0

For the first one, note that the denominator (x²) approaches zero as x→0. If the numerator (f(x)) did not approach zero as x→0, the limit would either be ±∞, or else it would be undefined. Since it is a finite value, the numerator must indeed approach zero as x→0.

For the second, make the same observation, but this time consider the numerator to be f(x)/x and the denominator to be x.

Answer 2

This is not hard if we look systematically. There may be more than one solution but we can work out on one

f(x)/x^2 = 1 and x->0
so power of x in f(x) cannot be < 2
if we focus on integral power we get
x^2+ a1x^3+ ..... satisfies the solution

devide by x^2
we get 1+ a1x+ a2x^2 tends to 1 which meets criteria

lim x->0 f(x) = 0
now f(x)/x = x+ a1x^2 -...
limit = 0 as x -.0
but there is a simpler method as well

lim x->0 h(x)g(x) = lim x->o h(x) lim x->o g(x) if both are defined
if h(x) = f(x)/x^2 and g(x) =x^2 we get x that is 0 f(x)
if h(x) = f(x)/x^2 g(x) = x we get 0 that is f(x)/x

Answer 3

Both limits are 0, as
f(x) = (f(x)/x^2) *(x^2) ---> 1*0=0
and f(x)/x = (f(x)/x^2) * x ---> 1*0=0