The Squeeze Theorem
The sandwich (aka squeeze) theorem
is very useful for computing limits like lim that would be difficult to evaluate
otherwishe. It says that if g(x) is sandwiched between
f(x) and h(x), and f(x) and h(x) have the same limit L as
x \to a, then g(x) also approaches L as x\to a:
Theorem:
Suppose that f(x) \le g(x) \le h(x) for all x that are
close to (but not equal to) a, and that
\displaystyle\lim_{x \to a} f(x) = \lim_{x \to a}
h(x) = L. Then \displaystyle\lim_{x \to a} g(x) =
L. |
When x is close to a, f(x) and h(x) are
close to L. So g(x) is somewhere between (a number close to
L)
and (another number close to L). This means that g(x)
must be
close to L!
Example: Evaluate \displaystyle{\lim_{x \to 0}
x^2 \sin\left(1/x\right)}.
DO: Why is it true that
-1\le \sin\theta\le 1 no matter what \theta is?
Solution: Since -1 \le \sin(1/x)\le 1, we can multiply
all three sides by the (\ge 0) value x^2, getting -x^2 \le
x^2 \sin(1/x)
\le x^2.
so g(x)=x^2 \sin(1/x) is squeezed
(or sandwiched)
between f(x)=-x^2 and h(x)=x^2. Since both -x^2 and x^2
approach 0 as x \to 0, we must also have \lim_{x \to 0} x^2
\sin(1/x) = 0.
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