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Say we know that a sequence $x_t$ converges to $x$ at the rate $O(1/log t)$. Can I say at what rate $\exp(x_t)$ converges to $\exp(x)$ ?

It comes as a subproblem in my work, and honestly I have no idea how to proceed or where to look. The convergence criteria which I am getting from textbooks are not helpful. How do I solve this? and how do I become better at handling questions like these ?

chillar007
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    Hint: use the derivative of the transformation at the limit. – Ian Feb 07 '24 at 17:18
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    Please Provide more information about your research and attempts at solving the problem. Also please refer to mathjax for writing mathematical expressions https://math.stackexchange.com/help/notation – Sarban Bhattacharya Feb 07 '24 at 17:19
  • Based on Ian's reply:

    $$| x_t - x | \sim O(\frac{1}{\log t}) \implies | f(x_t) - f(x) | \sim |\nabla f(x)| | x_t - x| \sim |\nabla f(x)| | O(1/\log t) \implies | f(x_t) - f(x) | \sim O( \frac{1}{ \log t}) $$

    Is this correct? So, as long as the derivative at the limit is bounded, the rate is the same?? that seems wrong. Shouldnt the transformation lead to slower or faster rates?

     – chillar007 Feb 08 '24 at 18:57
  • Nope, it's just a constant factor, unless the transformation is not differentiable near the limit. – Ian Feb 09 '24 at 01:49

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