How do I solve lim((3x^3 - 5x +2)/(4x^2 + 3)) as x approaches infinity?
I divided everything by the largest power of x, but I ended up getting a denominator of 0. Do I have to factor this?
(answer is infinity)
You did fine.
When you get c/0, where c is non-zero,
then the limit will approach infinity
It might be easier to see if you divide everything by only x^2.
then lim((3x^3 - 5x +2)/(4x^2 + 3))
= lim (3x - 5/x + 2/x^2)/(4 + 3/x^2)
so as x approaches infinity you are left with 3x/4.
Now as x ---> ∞ the numerator 3x ---> ∞
and thus 3x/4 ----> infinity.
Oh ok, thank you! My teacher taught us to divide by the largest power of x, so I didn't think of doing it like that.
But I don't quite get how c/0 is infinity. Isn't anything over 0 undefined?
To solve the limit lim((3x^3 - 5x +2)/(4x^2 + 3)) as x approaches infinity, you can use the concept of polynomial division or long division to simplify the expression.
First, divide the highest power of x from both the numerator and denominator. In this case, the highest power is x^3:
(3x^3 - 5x + 2) / (4x^2 + 3) = x^3 * (3 - 5/x^2 + 2/x^3) / (x^2 * (4 + 3/x^2))
As x approaches infinity, the terms with powers of x that have a finite limit tend to zero. Hence, the terms 5/x^2 and 2/x^3 will approach zero, and 3/x^2 becomes negligible.
Simplifying further, we get:
lim((3 - 5/x^2 + 2/x^3) / (4 + 3/x^2)) as x approaches infinity
Now, as x approaches infinity, both 5/x^2 and 2/x^3 will tend towards zero while 3/x^2 becomes negligible. Therefore, we can ignore these terms.
Simplifying further, we get:
lim((3) / (4)) as x approaches infinity
Since the limit is now a constant, the answer is simply 3/4.
Therefore, the correct answer is 3/4, not infinity.
It seems like there might be an error in your steps or calculations. Please double-check your work to ensure accuracy.