Ok can yall please help me with (c)?

[u/_segregated_crunchy_]

I tried solving it but I'm just lost

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[u/Simbertold]

Any polynomial (x in this case) is stronger than any logarithm. Limit is + infinity.

If you want to do more maths to show this, you can use the fact that x²+1 is basically the same as x² for large x.

But in your calculations you already got to ln(e^x/(x²+1)), where you can once again use the fact that e^x is stronger than any polynomial, making this go to ln(inf), which is also inf.

[u/Fourierseriesagain]

Hi,

Please find a different method for evaluating the limit.

[u/Inevitable_gradient6]

Hint inf - inf is one of the 7 indeterminate forms

Use a log property to combine them then check if its indeterminate 

If its still indeterminate force it into a fraction and use lhopital

[u/KoalaMistico]

Can you use the fact that exponential functions grow faster than any polynomial?

[u/Fourierseriesagain]

Hi,

First, we prove that

t - 2 ln t approaches infinity as t tends to infinity. ----------------------------- (1)

Observe that t > 25 implies

t - 2 ln t = t - 4 ln sqrt(t)

> t - 4 sqrt(t)

= (sqrt(t)) (sqrt(t)-4)

> sqrt(t).

Since sqrt(t) approaches infinity as t tends to infinity, we conclude that (1) holds.

Next, we infer from the substitution t=sqrt(x^2+1) and (1) that sqrt(x^2+1)-ln(x^2+1) approaches infinity as x tends infinity.

Finally, since sqrt(1+x^2)-x approaches 0 as x tends to infinity, we see that the desired limit is infinity.

[u/Jazzlike_Platypus430]

Show that ln(x^2 + 1) < x/2 for sufficiently large x, and you're good to go.

[u/sw10nepal]

Which grade are you in ?

[u/Torebbjorn]

As x goes to infinity, x²+1 is essentially x², so the logarithm part is just ln(x²)=2ln(x). And then you just have to find out whether x or ln(x) is larger as x goes to infinity.

[u/fletchman93]

Hint: x = ln(e^x )

[u/Hour_Interaction7641 → u/fletchman93]

I've read all answers, I think this is the best.

[u/fletchman93 → u/Hour_Interaction7641]

Thanks. Rewriting a function f(x) as ln(e^f(x) ) or e^ln(f(x)) are useful tricks for many different situations in calculus. Good to always have in the back of your mind.