lim as x --> infinity
( 4x + (3 / x^2) )
My work:
(4x^3 / x^2) + (3 / x^2)
(4x^3 + 3) / x^2
degree of top > degree of bottom
No horizontal asymptote, so the limit does not exist
The correct answer is infinity. How?
Also:
lim x → 0 (-x^2 + 4) / (x^2 - 1)
I got -1, but the right answer is -4
#1 is correct, but could be done directly. 3/x^2 -> 0
so, the limit is just 4x, which -> ∞
For #2, no tricks are involved. x^2 -> 0, so
-(x^2-4)/(x^2-1) -> -(0-4)/(0-1) = -4
To find the limit as x approaches infinity for the expression (4x + (3 / x^2)), you correctly rewrote the expression as (4x^3 + 3) / x^2. Since the degree of the numerator (4x^3 + 3) is greater than the degree of the denominator (x^2), there is no horizontal asymptote.
However, the limit can still exist and be evaluated. To determine the limit as x approaches infinity, you need to consider the behavior of each term of the expression as x becomes very large.
The term 4x^3 will increase without bound as x gets larger, since it has a higher exponent of x.
The term 3/x^2 will approach 0 as x increases without bound because the denominator (x^2) grows much faster than the numerator (3).
Since the term 4x^3 grows without bound and the term 3/x^2 approaches 0, the overall expression (4x^3 + 3) / x^2 also grows without bound as x approaches infinity.
Therefore, the correct answer is indeed infinity.