1. ## Some limit questions

Let I be an open interval such that 4 ∈ I and let a function f be defined on
a set D = I\{4}. Evaluate $\lim_{x \to 4} f(x)$ , where x + 2 ≤ f(x) ≤ x^2 − 10 for all
x ∈ D.

2)How to use the squeeze theorem to show that $\lim_{x\to0^+\left(\sqrt{x}e^{sin\frac1x}\right)}= 0$

2. ## Re: Some limit questions

Originally Posted by Vinod
Let I be an open interval such that 4 ∈ I and let a function f be defined on
a set D = I\{4}. Evaluate $\lim_{x \to 4} f(x)$ , where x + 2 ≤ f(x) ≤ x^2 − 10 for all
x ∈ D.
It is pretty obvious, isn't it, that for x= 4, the given equality becomes $\displaystyle 6\le f(4)\le 6$. If you are concerned about the fact that 4 is NOT in D, don't be. As long as there are points in D arbitrarily close to 4 we can take the limit as x goes to 4.

2)How to use the squeeze theorem to show that $\lim_{x\to0^+\left(\sqrt{x}e^{sin\frac1x}\right)}= 0$
Use the fact that $\displaystyle e^x$ is an increasing function and that $\displaystyle sin(1/x)$ lies between -1 and 1.