JBD2
Sep20-09, 02:34 PM
1. The problem statement, all variables and given/known data
Let \[f(x)=\begin{cases}{}
\frac{6a(x^2+1)}{2x^2+1}+\frac{b\log \big((x+1)^4\big)}{x} &\mbox{if } x<0,\\
-2a-4 &\mbox{if } x=0, \\
\frac{a\sin x}{x}+b &\mbox{if } x>0.
\end{cases}\]
Find \(a\) and \(b\) such that \(f\) is continuous at \(x=0\).
2. Relevant equations
Left and Right Hand limits are equal, as well as limit laws (If you're helping with the question I assume you know what they are)
3. The attempt at a solution
If f is continuous at x=0, the limit as x approaches 0 from the left should be equal to the limit as x approaches 0 from the right. With that, I took the limit of the part of f where x<0 and set it equal to the limit of the part of f where x>0.
Using L'hopital's rule I computed: \lim_{x \to 0-}\frac{4b\log{x+1}}{x} and the result is 4b*log(e). If I set this equal to the right hand limit, I can calculate that b=\frac{-5a}{4\log{e}-1}. Now I am stuck and am not sure what to do. Thanks for the help.
Let \[f(x)=\begin{cases}{}
\frac{6a(x^2+1)}{2x^2+1}+\frac{b\log \big((x+1)^4\big)}{x} &\mbox{if } x<0,\\
-2a-4 &\mbox{if } x=0, \\
\frac{a\sin x}{x}+b &\mbox{if } x>0.
\end{cases}\]
Find \(a\) and \(b\) such that \(f\) is continuous at \(x=0\).
2. Relevant equations
Left and Right Hand limits are equal, as well as limit laws (If you're helping with the question I assume you know what they are)
3. The attempt at a solution
If f is continuous at x=0, the limit as x approaches 0 from the left should be equal to the limit as x approaches 0 from the right. With that, I took the limit of the part of f where x<0 and set it equal to the limit of the part of f where x>0.
Using L'hopital's rule I computed: \lim_{x \to 0-}\frac{4b\log{x+1}}{x} and the result is 4b*log(e). If I set this equal to the right hand limit, I can calculate that b=\frac{-5a}{4\log{e}-1}. Now I am stuck and am not sure what to do. Thanks for the help.