We are tasked with finding the limit of the following expression as $x \to 2$:

$\lim_{x \to 2} \frac{4f(x) - g(x)}{\left(f(x) + 8\right)^{-1}}$

Given: $f(x) = 3 \quad \text{and} \quad g(x) = -2 \quad \text{when} \quad x \to 2$

Step 1: Substituting the values of $f(x)$ and $g(x)$

Substitute $f(x) = 3$ and $g(x) = -2$ into the expression:

$4f(x) - g(x) = 4(3) - (-2) = 12 + 2 = 14$ $f(x) + 8 = 3 + 8 = 11$

Thus, the denominator becomes:

$\left(f(x) + 8\right)^{-1} = 11^{-1} = \frac{1}{11}$

Step 2: Final substitution

Now, substitute these values back into the original limit expression:

$\frac{4f(x) - g(x)}{\left(f(x) + 8\right)^{-1}} = \frac{14}{\frac{1}{11}} = 14 \times 11 = 154$

Conclusion:

The limit is:

$\lim_{x \to 2} \frac{4f(x) - g(x)}{\left(f(x) + 8\right)^{-1}} = 154$

Would you like a detailed breakdown of any of the steps?

5 Related Questions:

1. How does one solve limits that involve more complicated functions?
2. What is the significance of inverse functions in limit calculations?
3. How does substitution help in solving limit problems?
4. Can limits be solved using L'Hôpital's Rule when the direct substitution fails?
5. What happens if $f(x)$ or $g(x)$ has discontinuities at $x = 2$?

Tip:

Always check if you can directly substitute values in limit problems before trying more complex methods like L'Hôpital's Rule.