suppose that t and w are continous functions such that both of their lim as x go

Question

suppose that t and w are continous functions such that both of their lim as x goes to inifinity = 0.

Explain whether the following statements are true or false.

I don't know how to answer these questions and what counter examples to give.

Sup pose that t and w are continuous functions such that g statements is True or False. If it is true, explain why. If it is false, a) Suppose/t(x)d converges and t(x)w(x) 0 on the interval [7,0o). Then )dz must b) Supposet()d converges and 0 t(x) w(x) on the interval [7, oo). Then()dz must c) Suppose t(x)dz diverges and t(x) Sw(x) 30 on the interval [7,00). Then (x)da must also give a specific counterexample.2 7 also converge. 7 also converge. diverge. d) Suppose t(x)dz diverges and 0 2 t(r) 2 w(r) on the interval [7, 0o). Then u()da must also e) Suppose that /t(r)dr converges and that t(x) s0 for all z 27. Then t(x)sin (x)dz must also 7 7 diverge. 7 converge false. rce tcze) Is the Smaller function,this is not true

Explanation / Answer

a)
True
This is cuz t(x) <= 0 and it converges, say the answer is -15.
But we have w(x) >= t(x) but <= 0
Thus, w(x) will lie within [-15 , 0)], therefore convergent!

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b)
t > w
t < 0 and w < 0
Lets say the integral of t is : -15
We know that integral of w is less than -15, but this need not be convergent, cud also be divergent
FALSE

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c)
t < w < 0
t < w , w < 0 and t < 0
t diverges
We know that t < w, so w must diverge for sure
TRUE

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d)
0 > t > w
t < 0
w < 0
t > w
We know that t diverges
w < t and thus, w may diverge or converge
FALSE

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e)
t <= 0
integral of t from 7 ot inf converges

Also we know that by comparison
t(x) * sin^2(x) <= t(x)

t(x) converges but is less than ZERO
We know that t(x) * sin^2(x) is also less than 0
but it could also diverge

So, FALSE

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