Limit problem

For the first question, we have that

$(342^{x} + 343^{x})^{\frac{1}{x}} = \left(343^{x}\left(1 + \left(\dfrac{342}{343}\right)^{x}\right)\right)^{\frac{1}{x}} =$

$343 \times \left(1 + \left(\dfrac{342}{343}\right)^{x}\right)^{\frac{1}{x}}$ ,

since $(343^{x})^{\frac{1}{x}} = 343$ . So if the first expression has a limit then it will be the same as that for the last expression.

For the second question, yes, since $a^{x} \to 0$ for $0 \lt a \lt 1$ as $x \to \infty$ we have $(1 + a^{x})^{\frac{1}{x}}$ being of the form $1^{0}$ , which is not one of the indeterminate forms , so we can go ahead and conclude that the limit is $1^{0} = 1$ .

We can also look at it this way. If $y = (1 + a^{x})^{\frac{1}{x}}$ then $\ln(y) = \dfrac{\ln(1 + a^{x})}{x}$ , which as $x \to \infty$ goes to $\dfrac{\ln(1)}{x} \to 0$ , and thus $y \to e^{0} = 1$ .