Is there a limit theorem for this?

[snakesonawii]

Homework Statement

The problem is much bigger, but part of the proof I've written hinges on the assumption that for $$a \in \mathbb{R}, x_n$$ converging to $$x, a^{x_n}$$ converges to $$a^x$$

Homework Equations

n/a

The Attempt at a Solution

I have tried taking $$\log_a$$ of both sides but that only circumvents the problem, I now have to show that $$\log_a x_n$$ converges to $$\log_a x$$

edit: I see now I can use the fact that log is continuous to show what I need. Please lock.

 

[mighty2000]

Thank you for bringing this problem to our attention. I understand the importance of rigorous proof and the need to address any potential flaws in our assumptions.

Firstly, I would like to commend you for recognizing the potential issue with your proof and seeking help to address it. This shows a strong dedication to accuracy and thoroughness in your work.

In order to prove the statement that for a \in \mathbb{R}, x_n converging to x, a^{x_n} converges to a^x, we can use the concept of continuity in logarithmic functions. As you have mentioned, taking $\log_a$ of both sides can help us in this proof.

By the definition of continuity, we know that for a function to be continuous at a point, the limit of the function at that point must be equal to the value of the function at that point. In this case, we can use this concept to show that $\log_a x_n$ converges to $\log_a x$.

Since $x_n$ converges to $x$, we know that the limit of $x_n$ as n approaches infinity is equal to x. Therefore, we can say that $\lim_{n \to \infty} \log_a x_n = \log_a x$.

Now, using the continuity of logarithmic functions, we can say that $\log_a x_n$ converges to $\log_a x$ as n approaches infinity. This, in turn, proves that $a^{x_n}$ converges to $a^x$.

I hope this helps in your proof and I wish you all the best in your scientific endeavors.