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anonymous
 4 years ago
lim x→0 (x^2− 2 sin x)/x
What do?
I'm not a big fan of limits, so all help is appreciated.
anonymous
 4 years ago
lim x→0 (x^2− 2 sin x)/x What do? I'm not a big fan of limits, so all help is appreciated.

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anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0\[ \lim_{x\rightarrow 0} \frac{x^22\sin(x)}{x} = \lim_{x\rightarrow 0} x  2\cdot \lim_{x\rightarrow 0} \frac{\sin(x)}{x} \]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Now the first of the two limits on the right evaluates to 0, clearly.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0And for the second, you can use L'Hopitals Rule or the Taylor expansion of sine, as you wish.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0By L'Hopitals Rule you get \[ \lim_{x\rightarrow 0} \frac{\sin(x)}{x} = \lim_{x\rightarrow 0} \frac{(\sin(x))^\prime}{(x)^\prime} = \lim_{x\rightarrow 0} \frac{\cos(x)}{1} = 1 \] where the ' denotes derivative with respect to x and the last equality holds because \[\cos(0)=1 \]

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Putting it all together your limit evaluates to 2.

anonymous
 4 years ago
Best ResponseYou've already chosen the best response.0Thank you that actually clears a lot of this limit thing up.

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.0\[\lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1\] is one limit that should be memorized. To see why it is true just look at any undergradute text on calculus.

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.1\[\lim_{x \rightarrow 0}(\frac{x^2}{x}2 \cdot \frac{\sin(x)}{x})=\lim_{x \rightarrow 0}x2 \lim_{x \rightarrow 0}\frac{\sin(x)}{x}\] \[02(1)=2\]

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.1It is true by squeeze thm or sandwich thm

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.1whatever you want to call it

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.0I usually go by squeeze theorem myself :)

myininaya
 4 years ago
Best ResponseYou've already chosen the best response.1i like sandwiches though

Zarkon
 4 years ago
Best ResponseYou've already chosen the best response.0sandwiches are good...but i don't want to be thinking about food when I'm doing math
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