Question: top-left: lim
x→∞
center: (
4^x + 9^x
──────────
x
)^(1/x)
Math sentence: The graph shows the expression lim as x approaches ∞ of ((4^x + 9^x) / x)^(1/x).User:
1. **Problem statement:** Evaluate $$\lim_{x\to\infty}\left(\frac{4^x+9^x}{x}\right)^{1/x}$$.
2. **Use the dominant term rule.** As $x\to\infty$, the term $9^x$ grows much faster than $4^x$, so
$$4^x+9^x\sim 9^x.$$
That means the expression behaves like
$$\left(\frac{9^x}{x}\right)^{1/x}.$$
3. **Split the power.** Use $\left(\frac{a}{b}\right)^{1/x}=\frac{a^{1/x}}{b^{1/x}}$:
$$\left(\frac{9^x}{x}\right)^{1/x}=\frac{(9^x)^{1/x}}{x^{1/x}}.$$
4. **Simplify each part.**
$$\frac{(9^x)^{1/x}}{x^{1/x}}=\frac{9}{x^{1/x}}.$$
For the denominator, a standard limit is
$$\lim_{x\to\infty}x^{1/x}=1.$$
So the whole expression tends to
$$\frac{9}{1}=9.$$
5. **Final answer:**
$$\boxed{9}$$