lim x->oo (x^2)sin(1/x)

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- May 21st 2007, 06:47 PMjephl`hospital
lim x->oo (x^2)sin(1/x)

- May 21st 2007, 07:01 PMThePerfectHacker
Do not use it!

$\displaystyle -1 \leq \sin \frac{1}{x} \leq 1$ for all $\displaystyle x\not = 0$

Thus,

$\displaystyle -x^2 \leq x^2 \sin \frac{1}{x} \leq x^2 $

Now, $\displaystyle \lim_{x\to 0} -x^2 = \lim_{x\to 0}x^2 =0$

By squeeze theorem,

$\displaystyle \lim_{x\to 0} x^2 \sin \frac{1}{x} =0$ - May 21st 2007, 07:04 PMjeph
are you sure? o.o

the answer my prof provided was oo - May 21st 2007, 07:14 PMThePerfectHacker
I did it for $\displaystyle x\to 0$ instead of $\displaystyle x\to \infty$.

In that case,

$\displaystyle \lim_{x\to \infty} x^2\sin \frac{1}{x} = +\infty$

But we do not need to use that rule.

Instead for sufficiently large $\displaystyle x$ consider the reciprocal, i.e. $\displaystyle \frac{1}{x^2\sin \frac{1}{x}}$

We can think of this reciprocal function as a composition $\displaystyle f\circ g$ where $\displaystyle g = \frac{1}{x}$ and $\displaystyle f=\frac{x^2}{\sin x}$. But at $\displaystyle x=0$ we define $\displaystyle f$ to be zero. Why? Because $\displaystyle \lim_{x\to 0}f = \lim_{x\to 0} \frac{x^2}{\sin x} = \lim_{x\to 0} x \cdot \frac{x}{\sin x} = 0 \cdot 1 =0$. This makes $\displaystyle f$*continous*at zero.

Now $\displaystyle \lim_{x\to \infty} g = 0$ and $\displaystyle f$ is continous at 0. By the limit composition rule the limit $\displaystyle \lim_{x\to 0}f\circ g = f(0) = 0$.

We have shown that $\displaystyle \lim_{x\to +\infty} \frac{1}{x^2\sin \frac{1}{x}} =0$. Which implies that its reciprocal is $\displaystyle +\infty$ that is, $\displaystyle \lim_{x\to +\infty} x^2 \cdot \sin \frac{1}{x} = \infty$.