Question

Find the limits of \(q=(3 v+5) /(v+2)\) \((a)\) As \(v \rightarrow 0\) (b) As \(v \rightarrow 5\) (c) As \(v \rightarrow-1\)

Short answer

Limits are \(\frac{5}{2}, \frac{20}{7}, 2\) for \(v \to 0, 5, -1\) respectively.

Step-by-step solution

Finding the Limit as v Approaches 0

Substitute \(v = 0\) into the function \(q = \frac{3v + 5}{v+2}\) to find the limit as \(v\) approaches 0. \[ \frac{3(0) + 5}{0 + 2} = \frac{5}{2} \] Thus, \(\lim_{{v \to 0}} q = \frac{5}{2}\).

Finding the Limit as v Approaches 5

Substitute \(v = 5\) into the function \(q = \frac{3v + 5}{v+2}\) to determine the limit as \(v\) approaches 5. \[ \frac{3(5) + 5}{5 + 2} = \frac{20}{7} \] Thus, \(\lim_{{v \to 5}} q = \frac{20}{7}\).

Finding the Limit as v Approaches -1

Substitute \(v = -1\) into the function \(q = \frac{3v + 5}{v+2}\) to calculate the limit as \(v\) approaches -1. \[ \frac{3(-1) + 5}{-1 + 2} = \frac{2}{1} = 2 \] Hence, \(\lim_{{v \to -1}} q = 2\).

Key concepts

Understanding Rational Functions

Rational functions are a central part of calculus and are simply functions that are the ratio of two polynomials. For example, given the function \( q = \frac{3v + 5}{v+2} \), the numerator is \(3v + 5\) and the denominator is \(v + 2\).
Rational functions are versatile and appear frequently in real-world modeling, particularly because they can represent relationships that involve division, such as speed or concentration.
When working with rational functions, it's crucial to understand the behavior of both the numerator and the denominator. If the denominator can be zero, it can cause the function to become undefined, leading to interesting properties such as vertical asymptotes. Additionally, by understanding the degrees of the polynomials, you can infer about their horizontal asymptotes. This makes rational functions an exciting study area, revealing plenty about function behavior as variables change.

The Process of Limit Evaluation

Limit evaluation is an essential topic in calculus used to understand the behavior of functions as the input approaches a particular value. In our example function, \( q = \frac{3v + 5}{v+2} \), evaluating limits allows us to determine how \(q\) behaves as \(v\) gets closer to certain values like 0, 5, or -1.
The process involves direct substitution of the approaching value into the function, and if it results in a well-defined number, that number is the limit. For instance, as \(v\) approaches 0, substituting 0 in for \(v\) leads to \(\frac{5}{2}\), thus \(\lim_{{v \to 0}} q = \frac{5}{2}\).
However, there can be cases where substitution gives a 0 in the denominator. This usually makes direct calculation not possible, and further methods like algebraic manipulation or L'Hôpital's Rule might be necessary. Understanding limit evaluation provides deep insight into how functions behave at points of discontinuity or where they become unbounded.

Approaching Values in Functions

When we talk about limits, we are exploring how the value of a function behaves as the input approaches certain specific points. In rational functions like \( q = \frac{3v + 5}{v+2} \), approaching values mean seeing what happens to \(q\) as \(v\) nears critical numbers like in our example: 0, 5, and -1.
The concept behind approaching values is simple but profound: knowing the behavior of a function around these key points can often tell us a lot about the overall behavior of the function itself. For example, as \(v\) approaches \(5\), the function \(q\) approaches \(\frac{20}{7}\), giving a precise value of what happens right at that point, assuming continuity.
If the value of a function can't be directly calculated because of an undefined point (like with division by zero), limits help bridge this gap, allowing us to make meaningful conclusions about function behavior. By examining the limits, thus approaching these values through different paths, you gain a thorough understanding of continuity and the possible occurrence of asymptotic behaviors.