To find the horizontal and vertical asymptotes of the function $f(x) = \frac{x-4}{x+3}$, follow these steps:

1. Vertical Asymptotes

The vertical asymptote occurs when the denominator equals zero, as the function becomes undefined.

$x + 3 = 0 \implies x = -3$

Thus, the vertical asymptote is:

$x = -3$

2. Horizontal Asymptotes

The horizontal asymptote is determined by comparing the degrees of the numerator and denominator:

• Both the numerator ($x-4$) and denominator ($x+3$) are of degree 1.
• When the degrees are the same, the horizontal asymptote is the ratio of the leading coefficients.

The leading coefficients are $1$ (from $x$) in both the numerator and denominator, so:

$y = \frac{1}{1} = 1$

Thus, the horizontal asymptote is:

$y = 1$

Final Answer:

• Vertical asymptote: $x = -3$
• Horizontal asymptote: $y = 1$

Would you like a detailed explanation of why horizontal or vertical asymptotes are determined this way?

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Here are 5 related questions to expand your understanding:

1. How do you find slant (oblique) asymptotes if the numerator's degree is higher than the denominator's?
2. What happens to the function $f(x) = \frac{x-4}{x+3}$ as $x \to \infty$ and $x \to -\infty$?
3. Can a function have more than one horizontal or vertical asymptote?
4. What is the significance of asymptotes in graphing rational functions?
5. How do you find the intercepts of the function $f(x) = \frac{x-4}{x+3}$?

Tip: To sketch a rational function, always identify its asymptotes and intercepts first!