Squeeze Theorem: Bounds for Limits, Sequences, and Multivariable Functions

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State the squeeze theorem.

If $g(x)\leq f(x)\leq h(x)$ near $a$ and both $g$ and $h$ tend to the same value $L$ as $x\to a$, then $f(x)$ tends to $L$ as well.

What are the lower and upper bounds in the theorem?

They are functions $g$ and $h$ that satisfy $g(x)\leq f(x)\leq h(x)$ near the point of interest. Thus, $g$ is the lower bound and $h$ is the upper bound.

Why is a common limit for the bounds essential?

The bounds force the middle function toward a single value when both outer functions approach that same value. If their limits differ, this theorem alone gives no definite limit for the middle function.

Does the squeeze theorem require $f(a)=L$?

No. It concerns values of $f(x)$ near $a$ and does not require the function’s value at $a$ to equal the limit—or even to be defined.

How does the $\varepsilon$–$\delta$ proof work?

Choose neighborhoods where the lower bound is greater than $L-\varepsilon$ and the upper bound is less than $L+\varepsilon$. Taking the smaller neighborhood size ensures both inequalities hold, squeezing $f(x)$ within $\varepsilon$ of $L$.

Quiz questions

1. If sequences $(a_n)$ and $(c_n)$ both converge to $\ell$, and $a_n\leq b_n\leq c_n$ for all sufficiently large $n$, what follows?Answer: The sequence $(b_n)$ converges to $\ell$.
2. To use the squeeze theorem for a multivariable limit, where must the bounding inequalities hold?Answer: At all points sufficiently near the target point.
3. If a function has the same limiting value along several paths to a point, what can be concluded about its full multivariable limit?Answer: Those matching path limits alone do not prove that the full limit exists.
4. For which angle measure does $\lim_{x\to0}\frac{\sin x}{x}=1$ hold?Answer: Radians.
5. A geometric squeeze gives the derivative of $\tan\theta$. What derivative results?Answer: $\sec^2\theta$.