I'm trying to solve the limit of this inequality. The question goes as follows:
If $$4x – 9 \leq f(x) \leq x^2 – 4x + 7$$ for $x \geq 0$, find $\lim_{x\to 4} f(x)$.
I'm not really sure how to go about this problem. I tried solving it using one-sided limits and got my answer as 7 but I'm not sure how to elegantly present my answer. I want to know how to present my answer in an appropriate mathematical way.
But I'm not sure if there is a way to solve it using Sandwich theorem. I would like to know if it's possible, as well.
Thanks.
Best Answer
Using the Sandwich Theorem, we have $\lim_{x \to 4} 4x-9 = 7$ and $\lim_{x \to 4} x^2-4x+7 = 7$, and since $f(x)$ is sandwiched in between, $\lim_{x \to 4} f(x) = 7$.