Problem 24 - Calculus: Early Transcendentals

Question

Compute the following limits. $$ \lim _{x \rightarrow 0} \frac{3 x^{2}+x+2}{x-4} $$

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2}\).

Step 1: Analyze the limit form

First, substitute \( x = 0 \) in the expression \( \frac{3x^2 + x + 2}{x - 4} \). This results in \( \frac{3(0)^2 + 0 + 2}{0 - 4} = \frac{2}{-4} = -\frac{1}{2} \). Since substituting \( x = 0 \) gives a valid numeric result, the limit can be directly evaluated without indeterminate forms or needing further simplification.

Step 2: Conclude the limit

Since substituting directly into \( x = 0 \) yields a definite number, the limit of \( \lim _{x \rightarrow 0} \frac{3x^2 + x + 2}{x - 4} \) is simply \( -\frac{1}{2} \). This shows the function is continuous at \( x = 0 \) in the context of the given limit.

Key Concepts

Direct Substitution Method

The Direct Substitution Method is often one of the quickest ways to evaluate limits in calculus. It involves replacing the variable in a function's limit with the value it's approaching and checking if the function results in a real number. If a real and finite value is obtained, then that value is the limit. This method works particularly well for rational functions when they are not resulting in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In the exercise provided, we used the Direct Substitution Method on the rational function \( \frac{3x^2 + x + 2}{x - 4} \) by substituting \( x = 0 \). Doing so yielded the value \( -\frac{1}{2} \), which means the limit exists and is defined. Importantly, this direct approach also implies there are no discontinuities right at this point that would complicate the limit result.

Rational Functions

Rational functions are quotients of two polynomials. A rational function \( f(x) = \frac{P(x)}{Q(x)} \) can introduce complexities when calculating limits, often due to the presence of zeros in the denominator, which can lead to undefined values.

These types of functions can exhibit behaviors that cause limits to not exist at certain points, like vertical asymptotes, or result in indeterminate forms which need further exploration using algebraic manipulation, cancellations, or other methods. However, if the polynomial in the denominator does not reach zero at the point we approach, or it evaluates to some finite real number, the limit can often be found through simple substitution.

This is the case with our exercise's function, \( \frac{3x^2 + x + 2}{x - 4} \), where substituting \( x = 0 \) into the denominator yields \( -4 \), a non-zero and finite result. Thus, this rational function allows the direct evaluation of the limit, leading directly to a finite result without encountering indeterminate or undefined values.

Continuity of Functions

Continuity is a crucial property in calculus that describes how a function behaves at a particular point and overall. A function is continuous at a point \( c \) if three conditions are met: the function is defined at \( c \), the limit as \( x \) approaches \( c \) exists, and the limit equals the function's value at \( c \).

For rational functions, continuity at a given point means that substituting the point into both the numerator and the denominator should not result in undefined expressions. In our exercise, the function \( \frac{3x^2 + x + 2}{x - 4} \) remains continuous at \( x = 0 \) because the substitution does not cause any division by zero or other discontinuities.

This allows us to find the limit using direct substitution without encountering discontinuous behavior, which means that the function behaves smoothly at this point. The resulting value from the direct substitution also highlights continuity in the sense that the function’s limit matches the expected output value.