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# Limit question

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1. why is lim (x->0) of (pi*x) / (sin(pi*x)) = 1?????
2. (Original post by eftio.gea)
why is lim (x->0) of (pi*x) / (sin(pi*x)) = 1?????
The Maclaurin series for sin(z)=z-z^3/(3!)+...
let z = pi*x
the lim as x tends to 0 lets you ignore the series except the first term
3. Three main ways of doing this: Taylor series, L'Hopital's rule and geometric.

For the last way, can you try to prove by use of a diagram that

for . The areas of a triangle and sectors will be helpful here. Then manipulate it to get the limit of .
4. I always heard the sky is the limit.
5. This is a bit fiddly. It depends on the way you define your trigonometric functions.

For example, both L'Hopitals Rule and the Taylor series suggested above requires that you know how to differentiate , but knowing how to differentiate means you need to know that anyway, so it's all circular (depending on your starting definitions).
6. (Original post by Alex:)
Three main ways of doing this: Taylor series, L'Hopital's rule and geometric.

For the last way, can you try to prove by use of a diagram that

for . The areas of a triangle and sectors will be helpful here. Then manipulate it to get the limit of .
For the limit to exist it has to exist from the positive and negative direction so you have to consider both 1st and 4th quadrant.

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