Exercise 4.2.11

Question

[Squeeze Theorem]
    Let $f, g$, and $h$ satisfy $f(x) \leq g(x) \leq$ $h(x)$ for all $x$ in some common domain $A$. If $\lim _{x \rightarrow c} f(x)=L$ and $\lim _{x \rightarrow c} h(x)=$ $L$ at some limit point $c$ of $A$, show $\lim _{x \rightarrow c} g(x)=L$ as well.

Ulisse Mini · Accepted Answer

$$\begin{aligned}
|g(x) - L| &\leq |g(x) - f(x)| + |f(x) - L| \
        &\leq |h(x) - f(x)| + |f(x) - L| \
        & \leq |h(x) - L| + |L - f(x)| + |f(x) - L| = |h(x) - L| +2|f(x) - L|
    \end{aligned}
    $$
    For a given $\epsilon$ we can find $\delta> 0$ so that $|h(x) - L| < \epsilon/3$ and $|f(x) - L| < \epsilon/3$, hence $\lim _{x \rightarrow c} g(x)=L$.

Exercise 4.2.11

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Exercise 4.2.11

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