Find the Limits

[nycmathguy]

Calculus
Section 2.3

Note: The following question is not covered by Jenn in any of her limits video lessons. However, I am curious about how this one is done.

Note: This is tricky to me. The limit is given. Must I multiply both sides by x^2 to do away with the fraction on the left side as step 1?

 

[MathLover1]

a.
if lim(f(x)/x^2),x->0)=5

then lim(f(x),x->0)/lim(x^2,x->0)=5

=>lim(f(x),x->0)=5lim(x^2,x->0)

=> f(0)=5*0^2
=> f(0)=0

b.

if lim(f(x)/x^2),x->0)=5, find if lim(f(x)/x),x->0)=5

write f(x)/x^2 as (f(x)/x)(1/x), then

lim((f(x)/x)(1/x),x->0)=5

lim((f(x)/x),x->0)*lim(1/x, x->0)=5

lim((f(x)/x),x->0)=5/lim(1/x, x->0)

=>f(0)=5/(1/0)=5*0=0

 

[nycmathguy]

a.
if lim(f(x)/x^2),x->0)=5

then lim(f(x),x->0)/lim(x^2,x->0)=5

=>lim(f(x),x->0)=5lim(x^2,x->0)

=> f(0)=5*0^2
=> f(0)=0

b.

if lim(f(x)/x^2),x->0)=5, find if lim(f(x)/x),x->0)=5

write f(x)/x^2 as (f(x)/x)(1/x), then

lim((f(x)/x)(1/x),x->0)=5

lim((f(x)/x),x->0)*lim(1/x, x->0)=5

lim((f(x)/x),x->0)=5/lim(1/x, x->0)

=>f(0)=5/(1/0)=5*0=0

I thank you very much for your help but I must say the following. As we move on through calculus, questions will be getting harder. As a result, I would prefer for you to use wolfram or LaTex for easy reading of your replies. Too many parentheses in your typing.