For f(x)=(3x^2-5x+2)/(2x^2-6) complete the statements.
f(x)->_________ as x->-infinity
f(x)->_________ as x->infinity
In the blanks, I say for the first one infinity then -infinity for the second one.
use an intuitive approach.
as x becomes "very large" your numerator is dominated by the 3x^2 term and your denominator is dominated by the 2x^2 term, the other terms become less and less significant as x gets bigger
so your function approaches f(x) = 3x^2/2x^2 = 3/2
so the limit is 3/2
To determine the limiting behavior of a rational function like f(x) = (3x^2 - 5x + 2)/(2x^2 - 6), you need to analyze the degrees and coefficients of the leading terms in the numerator and denominator polynomials.
1. As x approaches negative infinity (x -> -∞):
- Notice that both the numerator and denominator have a leading term of x^2.
- The coefficients of the leading terms in the numerator and denominator are positive (3 and 2, respectively).
- In this case, as x becomes extremely negative, the coefficient with the higher power of x dominates the behavior.
- Since the coefficient in the numerator is greater than the coefficient in the denominator, as x approaches negative infinity, f(x) will tend towards positive infinity.
- Therefore, you are correct in stating that f(x) -> infinity as x -> -∞.
2. As x approaches infinity (x -> ∞):
- Similar to the previous case, both the numerator and denominator have a leading term of x^2.
- The coefficients of the leading terms in the numerator and denominator are both positive (3 and 2).
- In this scenario, as x becomes extremely large, the behavior depends on the relative sizes of the coefficients.
- Since the coefficient in the numerator is greater than the coefficient in the denominator, as x approaches infinity, f(x) will again tend towards positive infinity.
- Therefore, you are also correct in stating that f(x) -> infinity as x -> ∞.
In summary:
- f(x) -> infinity as x -> -∞
- f(x) -> infinity as x -> ∞