RaySix

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Limit as x approaches infinity of a rational function.

Created by @nathanedwards

at October 31st 2023, 7:57:25 pm.

AP_Calculus_AB-Questions

#limits

#rational-functions

#limits-at-infinity

Question:

Find the limit as x approaches infinity of the function:

f(x) = \frac{3x^2 + 5x + 1}{2x^2 - 4x + 3}

Answer:

To find the limit as x approaches infinity, we need to determine the behavior of the function as x gets larger and larger.

We can start by dividing both the numerator and denominator by the highest power of x, which in this case is $x^2$ . This simplifies the function to:

f(x) = \frac{3 + \frac{5x}{x^2} + \frac{1}{x^2}}{2 - \frac{4x}{x^2} + \frac{3}{x^2}}

Taking the limit as x approaches infinity, we can see that the highest power of x in the numerator and denominator becomes negligible compared to the other terms. Therefore, we have:

\lim_{x \to \infty} f(x) = \frac{3 + 0 + 0}{2 - 0 + 0} = \frac{3}{2}

Thus, the limit as x approaches infinity of the function f(x) is 3/2.

Note: It is important to note that when finding the limit at infinity, we focus on the dominant terms (terms with the highest power of x) and ignore the lower power terms or constants. In this case, the dominant terms are the ones with powers of x, and dividing by the highest power of x allows us to isolate the dominant terms and determine their behavior as x approaches infinity.