Question

Need help solving this limit problem.

Answer 1

If
`f` and
`g` are continuous everywhere, that means that for any
`x=c`, the limit as
`x\to c` for either function is the value of that function at
`c`:

`\displaystyle\lim_(x\to c)f(x)=f(c)\,\text{and}\,\lim_(x\to c)g(x)=g(c)`

Applying some properties of limits, we can rewrite the original limit as

`\displaystyle\lim_(x\to2)(f(x)+4g(x))=\lim_(x\to2)f(x)+\lim_(x\to2)4g(x)=\lim_(x\to2)f(x)+4\lim_(x\to2)g(x)=16`

Given the continuity of
`f` and
`g`, we have

`f(2)+4g(2)=16\implies 4g(2)=16-3=13\implies g(2)=\frac{13}4`

So for both parts, the answer is 13/4.