Question

If $\underset{x\to a}{lim}\left[f\left(x\right)+2g\left(x\right)\right]=7$ and $\underset{x\to a}{lim}\left[4f\left(x\right)-g\left(x\right)\right]=1$ then find the value of $\underset{x\to a}{lim}\frac{6f\left(x\right)}{g\left(x\right)}$.

Answer 1

$\begin{array}{c}f\left(a\right)+2g\left(a\right)=7\to \left(i\right)\times 1\\ 4f\left(a\right)-g\left(a\right)=1\to \left(ii\right)2\\ f\left(a\right)+2g\left(a\right)=7\\ 8f\left(a\right)-2g\left(a\right)=2\\ \underset{–}{+-+}\\ 9f\left(a\right)=9\\ f\left(a\right)=1\\ f\left(a\right)+2g\left(a\right)=7\\ 2g\left(a\right)=7-1=6\\ g\left(a\right)=3\\ {lim}_{x\to a}\frac{6f\left(\pi \right)}{g\left(x\right)}\end{array}$

Applying limit

$\begin{array}{c}\frac{6f\left(a\right)}{g\left(a\right)}\\ \frac{6\times 1}{3}\\ =2\end{array}$