Taking the limit of this as s-> 0 (algebra based)

[rela]

Homework Statement

Hi all, this may sound trivial but I've been like cranking my brains on this. Probably I just can't see it. Will need someone to shed some light.

Taking the limit of [(1/s)(a-b)/(s^2 + cs + a)] as s-> 0

Homework Equations

Stated above

The Attempt at a Solution

By using L'Hopital rule, I know this limit tends to 0 itself. However, I just can't manipulate the expression algebraically to obtain the same limit of 0. I'm just unable to cancel out that 'solo' s at the denominator. Hence, it appears to me that the limit shoots up to infinity as s-> 0 through algebraic means. I know this is incorrect but I just couldn't figure it out.

Please kindly help me in guiding me along.

Thanks!

 

[Pere Callahan]

Homework Statement

Hi all, this may sound trivial but I've been like cranking my brains on this. Probably I just can't see it. Will need someone to shed some light.

Taking the limit of [(1/s)(a-b)/(s^2 + cs + a)] as s-> 0

Homework Equations

Stated above

The Attempt at a Solution

By using L'Hopital rule, I know this limit tends to 0 itself. However, I just can't manipulate the expression algebraically to obtain the same limit of 0. I'm just unable to cancel out that 'solo' s at the denominator. Hence, it appears to me that the limit shoots up to infinity as s-> 0 through algebraic means. I know this is incorrect but I just couldn't figure it out.

Please kindly help me in guiding me along.

Thanks!

You can't use l'Hopital's rule here, because the fraction is not of the form "0/0" or "inf/inf". I understadn what you wrote as
$$\lim_{s\to 0}\frac{\frac{1}{s}(a-b)}{s^2+cs+a}$$
The numerator tends to infinity, the denominator however tends to a, so yes, the expression "blows up" as s-->0.