Xác Định Giới Hạn Using Squeeze Theorem

Trap a wild function between two tame ones

• Some limits resist substitution and algebra — like $\displaystyle\lim_{x\to0}x^2\sin\tfrac1x$, which wiggles infinitely fast.
• The escape: trap the messy function between two friendlier ones.
• If the upper and lower "walls" both head to the same height, the trapped function has nowhere else to go.
• This is the Squeeze Theorem 夹逼定理 (also called the Sandwich Theorem).

What the theorem says

• Suppose near $c$ (except maybe at $c$ itself) we have $g(x)\le f(x)\le h(x)$.
• If $\displaystyle\lim_{x\to c}g(x)=\lim_{x\to c}h(x)=L$, then $\displaystyle\lim_{x\to c}f(x)=L$ too.
• The two bounds squeeze $f$ onto the single value $L$.
• You need two things: a valid inequality 不等式 near $c$, and equal limits for the bounds.

The Squeeze Theorem traps x squared sin(1/x) between minus x squared and x squared

The classic wiggle

• $\sin\tfrac1x$ swings between $-1$ and $1$ forever as $x\to0$ — no limit on its own.
• But multiply by $x^2$: since $-1\le\sin\tfrac1x\le1$, we get $-x^2\le x^2\sin\tfrac1x\le x^2$.
• Both walls $\pm x^2 \to 0$ as $x\to0$.
• Squeezed between them, $\displaystyle\lim_{x\to0}x^2\sin\tfrac1x=0$.

Checking the setup honestly

• The inequality only has to hold near $c$, not everywhere — a small interval around $c$ is enough.
• The bounds' limits must be equal; if the walls head to different heights, the theorem says nothing.
• Squeeze is the standard route to the famous $\displaystyle\lim_{x\to0}\frac{\sin x}{x}=1$, from the geometric bounds $\cos x \le \frac{\sin x}{x}\le 1$.
• Always name your two bounding functions explicitly — that is the heart of the argument.