Question:

Consider the function f(x) defined by

f(x) = (x^2 - 9) / (x - 3)

1. Evaluate the limit of f(x) as x approaches 3.

2. Find the vertical asymptote(s) of f(x).

3. Determine whether f(x) has a horizontal asymptote. If it does, find its equation. If it does not, explain why.

4. Determine whether f(x) has a removable discontinuity. If it does, find the value(s) of c such that f(x) = f(c) for all x ≠ c.

Answer:

1. To evaluate the limit of f(x) as x approaches 3, we can directly substitute x = 3 into the function:

f(3) = [(3^2) - 9] / (3 - 3)

The denominator becomes 0, which indicates potential discontinuity. However, we can simplify the expression by factoring the numerator:

f(3) = [(3 - 3)(3 + 3)] / 0

The numerator becomes 0 while the denominator remains 0. Therefore, we have an indeterminate form of 0/0, which requires further algebraic manipulation. Applying the limit laws, we can rewrite the expression:

f(3) = [(3)(6)] / 0

Here, we notice that the numerator is non-zero while the denominator is still 0. Hence, the limit of f(x) as x approaches 3 is undefined.

2. To find the vertical asymptote(s), we need to examine the behavior of f(x) as x approaches positive or negative infinity.

As x approaches infinity, we can evaluate the limit:

lim(x→∞) [(x^2 - 9) / (x - 3)] = lim(x→∞) [x^2 / x] = lim(x→∞) x = ∞

As x approaches negative infinity, we can evaluate the limit:

lim(x→-∞) [(x^2 - 9) / (x - 3)] = lim(x→-∞) [x^2 / x] = lim(x→-∞) x = -∞

From these calculations, we can observe that f(x) has no vertical asymptotes.

3. To determine whether f(x) has a horizontal asymptote, we need to analyze its behavior as x approaches positive or negative infinity.

As x approaches infinity, we can evaluate the limit:

lim(x→∞) [(x^2 - 9) / (x - 3)] = lim(x→∞) [x^2 / x] = lim(x→∞) x = ∞

As x approaches negative infinity, we can evaluate the limit:

lim(x→-∞) [(x^2 - 9) / (x - 3)] = lim(x→-∞) [x^2 / x] = lim(x→-∞) x = -∞

Since the limits as x approaches infinity and negative infinity yield non-vertical lines, f(x) does not have a horizontal asymptote.

4. To determine whether f(x) has a removable discontinuity, we need to check if there is a value of c such that f(x) = f(c) for all x ≠ c.

By simplifying the expression f(x) = (x^2 - 9) / (x - 3), we can rewrite it as:

f(x) = (x + 3), x ≠ 3

This simplified form represents a straight line with slope 1. Therefore, there is no value of c such that f(x) = f(c) for all x ≠ c, and thus, f(x) does not have a removable discontinuity.