How to deal with undefined limits?

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January 3rd 2011, 08:39 AM
crossbone

How to deal with undefined limits?

$\text{For}\displaystyle \lim_{x \to \infty} \frac{\sin{x}}{x}=\lim_{x \to \infty} \frac{1}{x}\times \lim_{x \to \infty} \sin{x}=0 \times \text{Unknown}$

Is the answer zero?
January 3rd 2011, 08:47 AM
Archie Meade

Quote:

Originally Posted by crossbone

$\text{For}\displaystyle \lim_{x \to \infty} \frac{\sin{x}}{x}=\lim_{x \to \infty} \frac{1}{x}\times \lim_{x \to \infty} \sin{x}=0 \times \text{Unknown}$

Is the answer zero?

$-1\ \le\ sinx\ \le\ 1$

$\displaystyle\lim_{x \to \infty}\ \left(-\frac{1}{x}\right)\ \le\ \lim_{x \to \infty}\frac{sin\;x}{x}\ \le\ \lim_{x \to \infty}\left(\frac{1}{x}\right)$
January 3rd 2011, 09:03 AM
crossbone

Thanks! just curious, what's the answer to $\displaystyle \lim_{x \to \infty} sin\;x \$ ? Undefined?
January 3rd 2011, 09:13 AM
Plato

Quote:

Originally Posted by crossbone

Thanks! just curious, what's the answer to $\displaystyle \lim_{x \to \infty} sin\;x \$ ? Undefined?

Yes that limit is not defined.
January 3rd 2011, 09:24 AM
crossbone

Thanks! btw, what are indeterminate forms as far as limits is concern?
$\frac{0}{0}, \frac{\infty}{\infty}, \frac{0}{\infty}, \frac{\infty}{0}, \infty \times 0$ ?
January 3rd 2011, 09:27 AM
Archie Meade

Quote:

Originally Posted by crossbone

Thanks! just curious, what's the answer to $\displaystyle \lim_{x \to \infty} sin\;x \$ ? Undefined?

Can you apply the squeeze theorem to $\displaystyle\lim_{x \to \infty}\frac{sin\;x}{x}\;\;?$

Edit: sorry, the absence of a denominator bypassed my vision!
Thanks Plato
(next post).
January 3rd 2011, 09:30 AM
Plato

Quote:

Originally Posted by Archie Meade

Can you apply the squeeze theorem to $\displaystyle\lim_{x \to \infty}\sin(x)?$

Not to that limit. It bounces from -1 to 1.

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