Questions: Evaluate the limit as x approaches plus or minus infinity of (3x^2 + 4x - 2) / (3x^2 - 2x + 2) and use these limits to determine the end behavior of f(x) = (3x^2 + 4x - 2) / (3x^2 - 2x + 2). Enter the exact answers. If there is no horizontal asymptote, enter NA. Limit as x approaches plus or minus infinity of (3x^2 + 4x - 2) / (3x^2 - 2x + 2) = Horizontal asymptote: y =

Evaluate the limit as x approaches plus or minus infinity of (3x^2 + 4x - 2) / (3x^2 - 2x + 2) and use these limits to determine the end behavior of f(x) = (3x^2 + 4x - 2) / (3x^2 - 2x + 2).

Enter the exact answers. If there is no horizontal asymptote, enter NA.

Limit as x approaches plus or minus infinity of (3x^2 + 4x - 2) / (3x^2 - 2x + 2) =

Horizontal asymptote: y =

Transcript text: Evaluate $\lim _{x \rightarrow \pm \infty} \frac{3 x^{2}+4 x-2}{3 x^{2}-2 x+2}$ and use these limits to determine the end behavior of $f(x)=\frac{3 x^{2}+4 x-2}{3 x^{2}-2 x+2}$. Enter the exact answers. If there is no horizontal asymptote, enter NA. \[ \lim _{x \rightarrow \pm \infty} \frac{3 x^{2}+4 x-2}{3 x^{2}-2 x+2}= \] Horizontal asymptote: $\boldsymbol{y}=$

Solution

Solution Steps

Step 1: Evaluate the Limit as $ x \to +\infty $

To find $ \lim_{x \to +\infty} \frac{3x^2 + 4x - 2}{3x^2 - 2x + 2} $, we observe that both the numerator and denominator are quadratic polynomials. The leading coefficients of both polynomials are $ 3 $. Therefore, the limit is given by the ratio of the leading coefficients: \[ \lim_{x \to +\infty} \frac{3x^2 + 4x - 2}{3x^2 - 2x + 2} = \frac{3}{3} = 1. \]

Step 2: Evaluate the Limit as $ x \to -\infty $

Next, we evaluate $ \lim_{x \to -\infty} \frac{3x^2 + 4x - 2}{3x^2 - 2x + 2} $. Similar to the previous limit, since both the numerator and denominator are quadratic polynomials with the same leading coefficient, we find: \[ \lim_{x \to -\infty} \frac{3x^2 + 4x - 2}{3x^2 - 2x + 2} = \frac{3}{3} = 1. \]

Step 3: Determine the Horizontal Asymptote

Since both limits as $ x \to +\infty $ and $ x \to -\infty $ are equal to $ 1 $, we conclude that the horizontal asymptote of the function $ f(x) = \frac{3x^2 + 4x - 2}{3x^2 - 2x + 2} $ is: \[ y = 1. \]