Question

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow 3^{-}} f(x)\) (b) \(\lim _{x \rightarrow 3^{+}} f(x)\) (c) \(\lim _{x \rightarrow 3} f(x)\) $$ f(x)=\frac{x^{2}-11 x+30}{x^{3}-4 x^{2}-3 x+18} $$

Short answer

\(\lim _{x \rightarrow 3^{-}} f(x) = -\infty\), \(\lim _{x \rightarrow 3^{+}} f(x) = \infty\), overall limit does not exist.

Step-by-step solution

Factor the Function

We start by factoring both the numerator and the denominator to simplify the expression. The numerator is a quadratic that factors as \((x-5)(x-6)\). The denominator is a cubic polynomial. After testing possible roots, it factors as \((x-3)(x-2)(x+3)\). Therefore, we have:\[ f(x) = \frac{(x-5)(x-6)}{(x-3)(x-2)(x+3)} \]

Approximate the Left Limit \\(x \rightarrow 3^{-}\\)

To find \(\lim _{x \rightarrow 3^{-}} f(x)\), we choose values slightly less than 3. For example, try \(x = 2.9\). Calculate:\[f(2.9) = \frac{(2.9-5)(2.9-6)}{(2.9-3)(2.9-2)(2.9+3)}\]Calculating both components gives a small negative numerator and a small negative denominator, which results in a positive value. Performing the computations gives a very large negative value, indicating \(\lim _{x \rightarrow 3^{-}} f(x) = -\infty\).

Approximate the Right Limit \\(x \rightarrow 3^{+}\\)

Now, for \(\lim _{x \rightarrow 3^{+}} f(x)\), choose values slightly greater than 3, such as \(x = 3.1\). Calculate:\[f(3.1) = \frac{(3.1-5)(3.1-6)}{(3.1-3)(3.1-2)(3.1+3)}\]The calculations result in a small negative numerator and a small positive denominator, leading to a large positive value. This suggests \(\lim _{x \rightarrow 3^{+}} f(x) = \infty\).

Determine the Overall Limit

The overall limit \(\lim _{x \rightarrow 3} f(x)\) exists only if both the left and right limits are equal. From the previous steps, we found:- \(\lim _{x \rightarrow 3^{-}} f(x) = -\infty\)- \(\lim _{x \rightarrow 3^{+}} f(x) = \infty\)Since these are not equal, \(\lim _{x \rightarrow 3} f(x)\) does not exist.

Key concepts

Numerical Approximation

To understand limits, we can use numerical approximation. This means guessing the limit by plugging in numbers close to the point we are interested in. Suppose we want to approximate \( \lim _{x \rightarrow 3^{-}} f(x) \). We can choose numbers slightly less than 3, such as 2.9 or 2.99. By substituting these values into the function, we can observe how it behaves as it gets closer to the value of 3. Let's see an example. For \( x = 2.9 \), we got a large negative result in the given problem because both the numerator and the denominator separately resulted in small negative values. This observation led us to conclude \( \lim _{x \rightarrow 3^{-}} f(x) = -\infty \). By doing similarly for the right-hand limit, we pick numbers slightly larger than 3, say 3.1 or 3.01, calculate \( f(x) \) at these points, to deduce the behavior of the limit from the right.In summary, numerical approximation involves using values approaching the point of interest from both sides and is a powerful tool when algebraic simplification alone isn't clear enough.

Factoring Polynomials

Factoring polynomials is a method that simplifies expressions by representing them as a product of simpler polynomials. In our exercise, this method helps us break down both the numerator and the denominator of the function \( f(x) \).Let's take a closer look. The numerator is a quadratic expression which factors into \( (x-5)(x-6) \). Similarly, the denominator is a cubic expression that factors into \( (x-3)(x-2)(x+3) \). The main goal of factoring is to rewrite the expression in a simpler form, sometimes revealing cancellations and simplifications. By canceling common factors, if any exist, the function simplifies, making it easier to handle and solve for limits and other calculations. But remember, it's crucial to factor correctly to ensure the function's value doesn't change. Factoring allows us to see potential behavior near certain values like \( x = 3 \) where the denominator could potentially become zero.

Left-Hand and Right-Hand Limits

Left-hand and right-hand limits help us examine how a function behaves as it approaches a specific point from the left and the right. In the context of limits, we often express this with notations like \( \lim_{x \to c^-} f(x) \) and \( \lim_{x \to c^+} f(x) \). For our specific problem, to find \( \lim _{x \rightarrow 3^{-}} f(x) \), we look at values approaching 3 from the left (values less than 3). In contrast, for \( \lim _{x \rightarrow 3^{+}} f(x) \), we consider values coming from the right (greater than 3).Often, a discrepancy between left-hand and right-hand limits indicates a discontinuity at the point, implying the overall limit does not exist at that point. This is exactly what happened in our exercise, where the left-hand limit was \(-\infty\) and the right-hand limit was \(\infty\), thus showing that \( \lim _{x \rightarrow 3} f(x) \) does not exist.

Infinite Limits

Infinite limits occur when the values of a function increase or decrease without bound as \( x \) approaches a specific number. In our exercise, we found that as \( x \) approaches 3 from the left, the value of \( f(x) \) became very large in the negative direction, resulting in an infinite limit, written as \(-\infty\). On the other hand, approaching 3 from the right made \( f(x) \) very large in the positive sense, written as \(\infty\). This behavior points to something very interesting in calculus: at certain points, functions can exhibit very different behaviors when approaching from opposite directions.Infinite limits often indicate a vertical asymptote at that point in the graph of the function. Here, the function \( f(x) \) does not "settle" into any particular value but instead grows larger and larger either in the positive or negative direction near that point.