fourier and laplace examples

PG1995 said:

Can't we do this limit (x->0){arctan(x/x)}? The limit evaluates to 45 degrees. If I can't do this then how can I sketch the spectrum?

If possible then please give these queries a look? Thanks a lot.

Regards
PG

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I don't think limits apply here, but even if you can formulate a functional form with a variable to take limits relative to, what would be the meaning of the number you get? It's like asking someone to travel in a northeast direction for a distance of 0 km. I can just as well tell them to go southwest for 0 km and they will still be standing in the same spot whether they follow your instructions, or mine. This is why the value is indeterminate. Vectors with no magnitude have no meaningful direction, or you could say that all angles are equally valid.

However, you can use a value of zero in plots, or put a different symbol to indicate an indeterminate value.

For your questions:

Q1, I don't know if it's correct because I didn't have time to check it.

Q2, The angle of zero seems correct for Bn=0, unless An is also zero (in which case it is indeterminate) , but even then, using zero for all cases (including indeterminate ones) should do no harm.

Q3 Yes, spectra are often shown even for DC, particularly for the amplitude spectra.

Thanks a lot, Steve.

Could you please help me with these queries Q1, Q2 and Q3? Thanks a lot.

Regards
PG

Q1: I'm not aware of a method. Personally, I just use intuition and trial-and-error.

Q2: You need to formally take the limit as the value goes to infinity. Keep in mind that exp(iw) as w goes to infinity, does not coverge to any finiite number, but it does oscillate and maintains a finite value (i.e. it does not go to infinity). However exp(w) goes to zero as w goes to infinity. These two facts help you get the right answer. exp(w+iw)=exp(w)exp(iw), and zero multiplied by a number that does not go to infinity is zero.

Q3: A quick glance and my recollection that 1-cos2x can be related to 2(sin(x))^2 tells me that your answer is probably correct and should match the book answer. However, this is a good example of how you should be able to check answers yourself. Even if you didn't see the trig identity and the way to formally show the equivalence. You can check spot values with a calculator or with Matlab. It's clear that they agree when w goes to zero (just take the limit). Then you can check other numbers like w=100 and w=pi and w=4.5676778888. The odds of having a wrong answer match at all these values is very very slim. Do you see how easliy engineers can double check? This is an important skill to learn.

However, being able to show it formally is important too and this is sometimes easy and sometimes hard. Sometimes it is easy and you just have a "brain-freeze" and dont' see the easy way. That's life.

Deleted by PG1995