Question

Taking the set of all finite real numbers as the domain of the function \(q=g(y)=v^{2}-\) \(5 v-2\) (a) Find the limit of \(q\) as \(v\) tends to \(N\) (a finite real number). (b) Check whether this timit is equal to \(g(N)\) (c) Check whether the function is continuous at \(N\) and continuous in its domain.

Short answer

(a) \( N^2 - 5N - 2 \); (b) Yes, the limit equals \( g(N) \); (c) The function is continuous at \( N \) and across its domain.

Step-by-step solution

Understanding the Function

The function given is a quadratic function in the form \( g(v) = v^2 - 5v - 2 \). This is a continuous and differentiable function over its entire domain, which consists of all real numbers.

Finding the Limit

To find the limit of \( q = g(v) \) as \( v \to N \), where \( N \) is a finite real number, we simply evaluate the function at \( N \). Limits of polynomial functions can be computed directly by substitution since they are continuous at all real points. Hence, \[ \lim_{{v \to N}} g(v) = g(N) = N^2 - 5N - 2. \]

Evaluating the Function at N

Now, we need to evaluate the function \( g(v) \) at \( v = N \). Substituting \( N \) into the function gives us: \[ g(N) = N^2 - 5N - 2. \]

Comparing the Limit and Function Value

From Steps 1 and 2, we have that both \( \lim_{{v \to N}} g(v) \) and \( g(N) \) evaluate to \( N^2 - 5N - 2 \). Therefore, the limit of the function as \( v \to N \) is equal to the value of the function at \( N \).

Checking Continuity at N

A function \( g(v) \) is continuous at a point \( N \) if \( \lim_{{v \to N}} g(v) = g(N) \). Since we found this equality to hold in Step 3, \( g(v) \) is continuous at \( N \).

Checking Continuity Across the Domain

Since \( g(v) \) is a polynomial function, it is continuous everywhere on its domain, which consists of all real numbers. Polynomial functions are continuous and differentiable across their entire domain.

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables raised to whole number powers. The function in our exercise, \( g(v) = v^2 - 5v - 2 \), is specifically a quadratic polynomial.

• The highest power of the variable, \( v \), is 2, making it a quadratic function.
• Polynomial functions are smooth and continuous, meaning there are no breaks or holes in their graphs.
• Such functions can be added, subtracted, and multiplied, and they always return a real number when given a real number input.

Given these properties, polynomial functions are very predictable and behave nicely over all real numbers. This is why evaluating limits is straightforward when dealing with polynomials.

Real Numbers

Real numbers include all the numbers you can find on the number line. This encompasses integers, fractions, and irrational numbers like \( \pi \) and \( \sqrt{2} \).

• Real numbers are infinite and uncountable, providing a thorough and complete set of values for evaluating functions.
• The domain of our function \( g(v) = v^2 - 5v - 2 \) is all real numbers, meaning you can plug any real number into this function and it will yield a result.
• Understanding that our domain consists of real numbers is crucial for solving limits and confirming the continuity of the function.

In essence, real numbers provide the flexibility needed to evaluate polynomial functions under any circumstances within the real number system.

Continuity at a Point

Continuity of a function at a specific point means that the function is unbroken or seamless at that point. If a function \( g(v) \) is continuous at \( v = N \), the limit of the function as \( v \) approaches \( N \) is equal to the function value at \( N \).

• The formal definition checks if \( \lim_{{v \to N}} g(v) = g(N) \).
• In our exercise, this condition holds true: \( \lim_{{v \to N}} g(v) \) is exactly \( g(N) \).
• This means that there is no sudden jump, gap, or hole at \( N \) for this function.

Thus, for the quadratic function \( g(v) = v^2 - 5v - 2 \), continuity at any point \( N \) is guaranteed.

Function Evaluation

Evaluating a function means calculating its value at a specific input. For polynomial functions, this typically involves substituting the input value into the function.

• In the exercise, substituting \( v = N \) into the function \( g(v) = v^2 - 5v - 2 \) gives us \( g(N) = N^2 - 5N - 2 \).
• This result can then be used to confirm the limit and continuity of the function at that point.
• Function evaluation is a simple yet powerful tool for confirming polynomial behaviors and properties.

Evaluating polynomial functions directly leads to insights about their continuity and limits, making such computations an essential skill in calculus and analysis.