Question: top-left: none
center: lim
x→∞
(
(4^x + 9^x)
──────────
x
)^1/x
top-right: none
bottom-left: none
bottom-right: none
it is over x
1. **Problem statement:** Evaluate $$\lim_{x\to\infty}\left(\frac{4^x+9^x}{x}\right)^{1/x}.$$
2. **Use the dominant term rule.**
For large $$x,$$ the bigger exponential dominates, so $$4^x+9^x\sim 9^x.$$
Thus,
$$\frac{4^x+9^x}{x}\sim \frac{9^x}{x}.$$
3. **Rewrite the expression.**
$$\left(\frac{4^x+9^x}{x}\right)^{1/x}\sim \left(\frac{9^x}{x}\right)^{1/x}.$$
Now split the power:
$$\left(\frac{9^x}{x}\right)^{1/x}=\frac{(9^x)^{1/x}}{x^{1/x}}.$$
4. **Simplify each part.**
$$\frac{(9^x)^{1/x}}{x^{1/x}}=\frac{9}{x^{1/x}}.$$
Also, $$\lim_{x\to\infty}x^{1/x}=1.$$
So the whole expression approaches
$$\frac{9}{1}=9.$$
5. **Final answer:** $$\boxed{9}$$