Questions: Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw f(x)=(x^2+2x-3)/(x^2-64) The function has no horizontal asymptote. The function has a slant asymptote.

Solution Steps

To solve the given problem, we need to find the intercepts, asymptotes, and analyze the behavior of the function $f(x) = \frac{x^2 + 2x - 3}{x^2 - 64}$.

1. Intercepts:

   • x-intercepts: Set the numerator equal to zero and solve for $x$.
   • y-intercept: Evaluate $f(0)$.

2. Vertical Asymptotes:

   • Set the denominator equal to zero and solve for $x$.

3. Horizontal/Slant Asymptotes:

   • Since the degrees of the numerator and denominator are the same, there is no horizontal asymptote.
   • To find the slant asymptote, perform polynomial long division on the numerator and denominator.

4. Behavior at Vertical Asymptotes:

   • Use limits to describe the behavior of the function as $x$ approaches the vertical asymptotes.

To find the x-intercepts, we set the numerator equal to zero: $$x^2 + 2x - 3 = 0$$ The solutions to this equation are: $$x = -3 \quad \text{and} \quad x = 1$$ Thus, the x-intercepts are $(-3, 0)$ and $(1, 0)$.

To find the y-intercept, we evaluate the function at $x = 0$: $$f(0) = \frac{0^2 + 2(0) - 3}{0^2 - 64} = \frac{-3}{-64} = \frac{3}{64}$$ Thus, the y-intercept is $(0, \frac{3}{64})$.

Vertical asymptotes occur where the denominator is equal to zero: $$x^2 - 64 = 0$$ Solving this gives: $$x = -8 \quad \text{and} \quad x = 8$$ Thus, the vertical asymptotes are $x = -8$ and $x = 8$.

Since the degrees of the numerator and denominator are the same, we perform polynomial long division: $$\text{Slant Asymptote: } y = 1$$

To analyze the behavior of the function at the vertical asymptotes, we take the limits: $$\lim_{x \to -8} f(x) = \text{zoo} \quad \text{and} \quad \lim_{x \to 8} f(x) = \text{zoo}$$ This indicates that the function approaches infinity as $x$ approaches both vertical asymptotes.

Final Answer