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.

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.

Transcript text: Find the intercepts and asymptotes, use limits to describe the behavior at the vertical asymptotes, and analyze and draw \[ f(x)=\frac{x^{2}+2 x-3}{x^{2}-64} \] The function has no horizontal asymptote. The function has a slant asymptote, $\square$ -

Solution

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} $.

Intercepts:
- x-intercepts: Set the numerator equal to zero and solve for $ x $.
- y-intercept: Evaluate $ f(0) $.
Vertical Asymptotes:
- Set the denominator equal to zero and solve for $ x $.
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.
Behavior at Vertical Asymptotes:
- Use limits to describe the behavior of the function as $ x $ approaches the vertical asymptotes.

Step 1: Finding the x-Intercepts

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) $.

Step 2: Finding the y-Intercept

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}) $.

Step 3: Finding the Vertical Asymptotes

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 $.

Step 4: Finding the Slant Asymptote

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

Step 5: Analyzing Behavior at Vertical Asymptotes

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

x-intercepts: $ (-3, 0) $ and $ (1, 0) $
y-intercept: $ (0, \frac{3}{64}) $
Vertical asymptotes: $ x = -8 $ and $ x = 8 $
Slant asymptote: $ y = 1 $
Behavior at vertical asymptotes: $ \lim_{x \to -8} f(x) = \text{zoo}, \lim_{x \to 8} f(x) = \text{zoo} $

Thus, the final boxed answer is: \[ \boxed{\text{x-intercepts: } (-3, 0), (1, 0); \text{ y-intercept: } (0, \frac{3}{64}); \text{ vertical asymptotes: } x = -8, 8; \text{ slant asymptote: } y = 1} \]

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