A few calc problems...?

Question

I finished most of my hw except these problems. Please show work.
1) Find the limit:
lim x->infinity (x^4 +x^3)/(12x^3+128)

2) Find the limit:
lim x -> negative infinity e^(3t)inversesine(1/t)

3) If lim x->c (f(x)+g(x)) = 3 and lim x->c (f(x)-g(x)) = -1, find lim x->c f(x)g(x).

3) Find the limit and state if the function is continuous at the point being approached.

lim x->0+ sin((pi/2)e^(square root x)

4) Find a simple basic function as a left end behavior model and a simple basic function as a right end behavior model.

y = (2x^4 - x^3 + 2^x - 1) / (2 - x)

thank you very much!

Answer 1

I will only be answering number 1, but I'm not really sure of what to do so I will give 2 approaches and get your opinion.

1. As x approches infinity, you will get infinity/infinity for the limit.

You find the derivative of top and bottom.

4x^3 + 3x^2 / 36x^2

From this point I suggest you cancel out some terms before doing the limit.

4/3x + 1 / 12

Plug in infinity for x.

Infinity is your answer.

Second approach or answer:

Differentiate again.

4x^3 + 3x^2 / 36x^2

Differentiate.

12x^2 + 6x / 72x

Find the derivative AGAIN.

24x + 6 / 72


Plug infinity for x.

And you'll get infinity again.

For Question # 2, there is no x so there is no answer.

For the rest not so sure.

Check with a better math pro to find an answer that is more likely to be correct.

The reason I differentiated both top and bottom is because it's the L'Hoptial's rule and you follow it when you get infinity/infinity or 0/0 when you plug in the value x is approaching.

If you happen to get infinity/infinity or 0/0 again and again, keep differentiating.

Never mind, I'll do #3 also.

3. Replace x with c

So basically, this means:

f(c)+g(c) = 3

f(c)-g(c) = -1

Use the bottom equation and add g(c)

f(c) = g(c)-1

This would mean g(c) = 2 because

(g(c)-1) + g(c) = 3

2g(c) = 4

g(c) = 2

f(c) = 1

f(c)g(c) = 2(1)

The product is 2.

Answer 2

1) Find the limit:
lim x→∞ of [x^4 + x^3)/(12x^3 +128)]

The principle is that the highest power of x dominates as x→∞, so the limit is ∞. To see this divide numerator and denominator by x^4.

lim x→∞ of [1 + 1/x)/(12/x + 128/x^4)] = ∞