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Math Help - limit as x -> infinity?

  1. #1
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    limit as x -> infinity?

    How does one show that the limit as x approaches ∞ x^2 sin \frac {1}{x} is equal to ∞ mathematically? My solution book claims it is just the limit as h approaches zero from the right \frac {1}{h} \frac {sin h}{h} is equal to ∞. How does the solution booklet workout? I don't see the logic in it.
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  2. #2
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    Re: limit as x -> infinity?

    Quote Originally Posted by Barthayn View Post
    How does one show that the limit as x approaches ∞ x^2 sin \frac {1}{x} is equal to ∞ mathematically? My solution book claims it is just the limit as h approaches zero from the right \frac {1}{h} \frac {sin h}{h} is equal to ∞. How does the solution booklet workout? I don't see the logic in it.
    \displaystyle \begin{align*} \lim_{x \to \infty}x^2\sin{\frac{1}{x}} &= \lim_{x \to \infty}x\cdot \frac{\sin{\frac{1}{x}}}{\frac{1}{x}} \end{align*}

    Now if you let \displaystyle h = \frac{1}{x} and note that as \displaystyle x \to \infty, h \to 0, then

    \displaystyle \begin{align*} \lim_{x \to \infty}x \cdot \frac{\sin{\frac{1}{x}}}{\frac{1}{x}} &= \lim_{h \to 0}\frac{1}{h} \cdot \frac{\sin{h}}{h} \\ &= \lim_{h \to 0}\frac{1}{h} \cdot \lim_{h \to 0}\frac{\sin{h}}{h} \\ &= \infty \cdot 1 \\ &= \infty \end{align*}
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    Re: limit as x -> infinity?

    Quote Originally Posted by Prove It View Post
    \displaystyle \begin{align*} \lim_{x \to \infty}x^2\sin{\frac{1}{x}} &= \lim_{x \to \infty}x\cdot \frac{\sin{\frac{1}{x}}}{\frac{1}{x}} \end{align*}

    Now if you let \displaystyle h = \frac{1}{x} and note that as \displaystyle x \to \infty, h \to 0, then

    \displaystyle \begin{align*} \lim_{x \to \infty}x \cdot \frac{\sin{\frac{1}{x}}}{\frac{1}{x}} &= \lim_{h \to 0}\frac{1}{h} \cdot \frac{\sin{h}}{h} \\ &= \lim_{h \to 0}\frac{1}{h} \cdot \lim_{h \to 0}\frac{\sin{h}}{h} \\ &= \infty \cdot 1 \\ &= \infty \end{align*}
    thanks I see it now.
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