- #1

Astudious

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1: [tex]\lim_{x\rightarrow0} (\frac{cosec(x)}{x^3} - \frac{sinh(x)}{x^5})[/tex]

I don't really know what to do with this one. I tried differentiating a few times but it doesn't seem to work. Rewrite cosec(x) as sin(x) and then Taylor series, maybe, but I can't get it to work usefully - which terms to neglect and why?

2: [tex]\lim_{x\rightarrow0} (\int_{x}^{\frac{\pi}{2}} (\frac{ycos(y)-sin(y)}{y^2}) dy)[/tex]

The solution equates this expression to

[tex]\lim_{x\rightarrow0} (\int_{x}^{\frac{\pi}{2}} (\frac{d}{dy} (\frac{sin(y)}{y})) dy)[/tex]

from which the value of the limit follows easily. But what are the steps to calling these expressions equal?

3: My textbook also notes that the limit is distributive over addition, it shows that lim(f(x)+g(x)) = lim(f(x)) + lim(g(x)) and lim(f(x)*g(x)) = lim(f(x)) * lim(g(x)). It then says that this does not hold if you end up with a 0/0 or infinity/infinity or infinity*0. But even with these exceptions the rules still seem sketchy - can't you end up with infinity-infinity, which is not a valid result? How should I modify the rules to show when it safe to split the limits like this, and when it is not?