please help me solve this calculus problem.
GOD bless thee abundantly.

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@yinkatolu from the first principle

Number 1:

Lim x->oo X√3/x^2

Solution:
Divide the numerator and denominator by the highest power of x, in this case x^2

=X√3/x^2/  x^2/x^2

=√3/x / 1
Remember as X tends to infinity, C/x tends to zero, where C is a constant.
= 0/1
=0.

Follow the same procedure for question 2 and 3. If done correctly you should get 2 for question 2, and 0 for question 3.

Question 4 is trivial, although it'll be long: just apply the first principle.
x= X + ∆x; y = Y + ∆y.
Substitute those for X and Y respectively so the question reads:
Y + ∆y = (∆x + x)^4 + 5(x + ∆x)^3 -4(x + ∆x)^2 +7(x + ∆x) -2
Expand this and make ∆y the subject if the formula (don't forget to substitute the original function for Y)
Divide through by ∆x, then limit ∆x to 0.
That's all to it.
I trust you can take it from here.

The answer should be:

4x^3 +15x^2 -8x + 7

Hope this helps.
Cheers.

PS: Try out different practice questions so you can master it.

Thanks so much guys.