Question

Given \(q=5-1 / v,(v \neq 0),\) find: (a) \(\lim _{x \rightarrow+\infty} 9\) (b) \(\lim _{n \rightarrow-\infty} 9\)

Short answer

Both limits evaluate to 9, since the limit of a constant is the constant itself.

Step-by-step solution

Understanding the Problem

We are given a mathematical expression and asked to evaluate two limits. These limits involve the number 9, which does not depend on the variable approaching infinity.

Analyzing part (a)

In part (a), we need to evaluate \[ \lim_{x \rightarrow +\infty} 9 \] Here, 9 is a constant and does not depend on \(x\). Regardless of the value of \(x\), even as \(x\) approaches positive infinity, the constant value remains unchanged.

Solving part (a)

The limit of a constant is the constant itself. Thus, \[ \lim_{x \rightarrow +\infty} 9 = 9 \]

Analyzing part (b)

In part (b), we need to evaluate \[ \lim_{n \rightarrow -\infty} 9 \] Similar to part (a), the number 9 is independent of the variable \(n\). As \(n\) approaches negative infinity, the value of the expression remains the constant 9.

Solving part (b)

Since we are dealing with a constant, the limit is simply the constant value. Thus, \[ \lim_{n \rightarrow -\infty} 9 = 9 \]

Key concepts

Constant Function in Limits

When dealing with limits in calculus, one important concept is the behavior of a constant function. A constant function is a function that always returns the same value regardless of the input. In mathematical terms, if you have a function defined as \( f(x) = c \), where \( c \) is a constant, the output remains the same no matter what value \( x \) takes.

This is why, in our problem, the limit of a constant number, like 9, as \( x \) approaches positive or negative infinity, remains the same. The presence of a constant value keeps it immune to changes in the approaching value—in this case, infinity.

Understanding Infinity in Mathematics

Infinity is a concept that often appears in calculus and represents the idea of something that is unbounded or limitless. It isn't a number in the traditional sense, but rather a way to describe something that grows indefinitely.

In the context of limits, we examine how functions behave as their inputs become very large (approaching \(+\infty\)) or very small, moving towards negative infinity (\(-\infty\)).

• When \( x \) approaches \(+\infty\): We look at how the values of a function behave when the input becomes very large in the positive direction.
• When \( n \) approaches \(-\infty\): We observe the function's behavior as the input becomes very large in the negative direction.

This concept is essential for understanding how functions and expressions behave at the extremes.

Interpreting Mathematical Expressions

Mathematical expressions are a combination of numbers, variables, operators, and mathematical functions that represent a specific value or relationship. In calculus, these expressions often include limits, derivatives, or integrals.

To solve a limit problem, it is crucial to interpret the given mathematical expression accurately. Knowing the type of expression, particularly whether it's constant or variable-dependent, can simplify the process of limits evaluation.

• For constant expressions like 9, the output remains unchanged as variables approach infinity.
• If an expression had variables dependent on \( x \) or \( n \), you'd need additional strategies to assess the limit.

Success in limits evaluation starts with a solid understanding of the expressions involved.

Evaluating Limits of Expressions

Limit evaluation is a foundational concept in calculus that deals with determining the behavior of functions as the input approaches a particular point. When evaluating limits, it is important to consider the type of expression and the behavior of the function near the point of interest.

With constant functions, such as the expression \( 9 \) in our example, the evaluation is straightforward:

• As \( x \) or \( n \) approaches any value, including infinity, the limit simply remains the constant value \( 9 \).

For more complicated expressions involving variables, the limit evaluation requires analyzing the function's behavior as it approaches critical points. This often involves algebraic manipulation and the application of limit rules and theorems. Understanding these concepts ensures accurate evaluation of limits.