Question

Find the limits of the function \(q=7-9 v+v^{2}:\) (a) As \(v \rightarrow 0\) (b) As \(v \rightarrow 3\) (c) As \(v \rightarrow-1\)

Short answer

The limits are 7, -11, and 17 for \( v \to 0 \), \( v \to 3 \), and \( v \to -1 \) respectively.

Step-by-step solution

Understand the function

The given quadratic function is \( q = 7 - 9v + v^2 \). We need to find the limits as \( v \) approaches different values. This involves substituting the values into the function and simplifying.

Evaluate limit as v approaches 0

Substitute \( v = 0 \) into the function: \[q(0) = 7 - 9 \times 0 + 0^2 = 7.\]So, the limit \( \lim_{{v \to 0}} q(v) = 7 \).

Evaluate limit as v approaches 3

Substitute \( v = 3 \) into the function: \[q(3) = 7 - 9 \times 3 + 3^2 = 7 - 27 + 9 = -11.\]So, the limit \( \lim_{{v \to 3}} q(v) = -11 \).

Evaluate limit as v approaches -1

Substitute \( v = -1 \) into the function: \[q(-1) = 7 - 9 \times (-1) + (-1)^2 = 7 + 9 + 1 = 17.\]So, the limit \( \lim_{{v \to -1}} q(v) = 17 \).

Key concepts

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a particular value. Essentially, limits tell us the value that a function approaches as the independent variable (in this case, \(v\)) gets closer and closer to a specific number. In our exercise, we're exploring how the quadratic function \(q = 7 - 9v + v^2\) behaves as \(v\) approaches 0, 3, and -1.
When evaluating limits, substituting the value directly into the function is a common method, provided the function is continuous at that point. In our example, for each of the specified values, substituting the number for \(v\) helps us determine the limit at those values as follows:

• For \(v \to 0\), substitute 0 into the function.
• For \(v \to 3\), substitute 3 into the function.
• For \(v \to -1\), substitute -1 into the function.

Integrating such techniques allows us to simplify and understand the behavior of more complex functions, which is essential for solving problems across various fields of mathematics and engineering.

Quadratic Functions

Quadratic functions are polynomials of degree 2, typically written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Our function \(q = 7 - 9v + v^2\) is a quadratic function. It has the standard form \(v^2 - 9v + 7\), where \(a = 1\), \(b = -9\), and \(c = 7\).
Quadratic functions often form a parabolic curve on a graph, opening either upwards or downwards depending on the sign of the coefficient \(a\). In our example, since \(a = 1\) is positive, the parabola opens upwards. The vertex of a parabola is its highest or lowest point, and it can be a key focus point when analyzing the function's graph.
Quadratics are particularly interesting because they have different properties depending on their discriminant, \(b^2 - 4ac\):

• If it's positive, the quadratic has two distinct real roots.
• If it's zero, there is exactly one real root.
• If it's negative, the roots are complex and not visible on the real-number graph.

Understanding these basics helps us evaluate the limits accurately and provides insight into the nature of solutions within quadratic functions.

Substitution Method

The substitution method is an incredibly useful technique when assessing limits, especially when dealing with polynomials like quadratic functions. It involves replacing the variable in the function with the specified value, which helps determine the function's behavior as the variable approaches a particular number.
In our exercise, we use substitution directly:

• For \(v \to 0\), substituting 0 gives us the limit \(7\).
• For \(v \to 3\), substituting 3 yields the limit \(-11\).
• For \(v \to -1\), substituting -1 results in the limit \(17\).

This process is generally straightforward when the function is continuous at the point of interest. If not, additional methods, such as factoring, might be required to simplify the function first. Substitution not only simplifies finding limits but also plays a crucial role in various calculus applications, like integration and differentiation, and aids in understanding the behavior of functions at specific points.