Question

For what values a and b is the limit as x approaches 0 of (a-cos(bx))/x^2 = 50

Answer 1

\[\lim_{x\to0}\frac{a-\cos bx}{x^2}=\frac{a-1}{0}\]
Clearly, this is undefined for \(a\not=1\). But since the limit must exist, we require that the numerator also approaches 0 so that we can apply L'Hopital's rule. This means we must have \(a=1\).

Applying L'Hopital's rule yields
\[\lim_{x\to0}\frac{b\sin bx}{2x}=\frac{0}{0}\]
Apply the rule again:
\[\lim_{x\to0}\frac{b^2\cos bx}{2}=\frac{b^2}{2}\]
The limit is known to be 50, so you must have
\[\frac{b^2}{2}=50\]

Answer 2

awesome, thank you

Answer 3

You're welcome