f(x) = (x + 2)/(x² - 4). Identify all vertical asymptotes, any holes in the graph, and show how the function behaves near each discontinuity.

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Let's analyze the rational function f(x) equals x plus 2 divided by x squared minus 4. First, we factor the denominator to get x plus 2 divided by the product of x minus 2 and x plus 2. Notice that x plus 2 appears in both numerator and denominator, so it cancels out. This gives us the simplified form: 1 divided by x minus 2, for all x not equal to negative 2. This simplification reveals two key points: a vertical asymptote at x equals 2, where the denominator becomes zero, and a hole at x equals negative 2, where the common factor canceled out. The y-coordinate of the hole is negative one-fourth, which we can verify by evaluating the limit as x approaches negative 2. Now let's examine how the function behaves near each discontinuity. Near the hole at x equals negative 2, the function approaches negative one-fourth from both sides. This is a removable discontinuity, where the function is well-behaved but simply undefined at that specific point. Near the vertical asymptote at x equals 2, the behavior is quite different. As x approaches 2 from the left, the function values decrease without bound, approaching negative infinity. This happens because the denominator x minus 2 approaches zero from the negative side. Conversely, as x approaches 2 from the right, the function values increase without bound, approaching positive infinity. This occurs because the denominator approaches zero from the positive side. This behavior creates the characteristic vertical asymptote shape in the graph. Let's verify our findings algebraically. For the hole at x equals negative 2, we start with the original function and factor the denominator. Since x plus 2 appears in both numerator and denominator, we can cancel it out, giving us 1 divided by x minus 2. Now we can evaluate this simplified expression as x approaches negative 2, which gives us 1 divided by negative 4, or negative one-fourth. For the vertical asymptote at x equals 2, we again simplify to 1 divided by x minus 2. As x approaches 2 from the left, x minus 2 is negative and approaches zero, so the function approaches negative infinity. As x approaches 2 from the right, x minus 2 is positive and approaches zero, so the function approaches positive infinity. The table of values confirms our analysis, showing how the function values grow extremely large in magnitude near x equals 2, and approach negative one-fourth near x equals negative 2. Let's summarize our analysis of the rational function f(x) equals x plus 2 divided by x squared minus 4. After factoring the denominator, we found that the function simplifies to 1 divided by x minus 2 for all x not equal to negative 2. This simplification revealed two key discontinuities: a vertical asymptote at x equals 2, and a removable discontinuity, or hole, at x equals negative 2. Near the vertical asymptote, the function approaches negative infinity as x approaches 2 from the left, and positive infinity as x approaches 2 from the right. At the hole, the function approaches negative one-fourth as x approaches negative 2 from either side. These behaviors are characteristic of rational functions where factors cancel between numerator and denominator, creating holes, and where remaining factors in the denominator create vertical asymptotes. Understanding these discontinuities is essential for graphing rational functions and analyzing their behavior. To summarize what we've learned about rational functions and discontinuities: First, rational functions can have vertical asymptotes at points where the denominator equals zero and the numerator doesn't. Second, holes occur at points where common factors cancel between the numerator and denominator. For our specific function, f(x) equals x plus 2 divided by x squared minus 4, we identified a vertical asymptote at x equals 2 and a hole at x equals negative 2 with y-value negative one-fourth. Near vertical asymptotes, function values approach infinity from opposite directions—negative infinity from one side and positive infinity from the other. Finally, algebraic simplification is a powerful tool that reveals the true behavior of rational functions, allowing us to identify and classify all discontinuities. These concepts are fundamental for understanding the behavior of rational functions in calculus and applied mathematics.