Problem 22 - Calculus for Biology and Medicine

Question

Use the formal definition of limits to prove each statement. \(\lim _{x \rightarrow c}(m x+b)=m c+b\), where \(m\) and \(b\) are constants

Step-by-Step Solution

Answer

The limit \( \lim _{x \rightarrow c}(m x+b)=m c+b \) is proven using the \( \epsilon-\delta \) definition of limits.

Step 1: Understand the Definition

The formal definition of a limit states that \( \lim_{{x \to c}} f(x) = L \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \). For this problem, \( f(x) = mx + b \) and \( L = mc + b \).

Step 2: Set Up the Inequality

Using the definition, we set up the inequality \( |(mx+b) - (mc+b)| < \epsilon \). Simplifying this, we get \( |mx - mc| < \epsilon \).

Step 3: Factor the Expression

Notice that \( |mx - mc| = |m(x - c)| \). We can further simplify this to \( |m| |x - c| \). Thus, the inequality becomes \( |m| |x - c| < \epsilon \).

Step 4: Choose Delta

To satisfy the inequality, we want \( |m| |x - c| < \epsilon \). If \( m eq 0 \), choose \( \delta = \frac{\epsilon}{|m|} \). This ensures that whenever \( |x - c| < \delta \), \( |m| |x - c| < \epsilon \).

Step 5: Conclusion

Since for every \( \epsilon > 0 \), we can choose \( \delta = \frac{\epsilon}{|m|} \) such that the inequality \( |mx + b - (mc + b)| < \epsilon \) is satisfied, we can conclude that \( \lim _{x \rightarrow c}(m x+b)=m c+b \) as required.

Key Concepts

Limits

Limits are a foundational concept in calculus, helping us understand behaviors of functions as inputs approach certain values. A limit describes what happens to a function's output as its input gets arbitrarily close to a certain point. For example, if the limit of function \( f(x) \) as \( x \) approaches \( c \) is \( L \), it means that as \( x \) gets very close to \( c \), \( f(x) \) gets very close to \( L \). Limits are essential because they provide a way to analyze and describe continuity, differentiability, and the behavior of functions at infinity.

• Exploring Continuity: Limits help us determine if a function is continuous at a point, which means there is no interruption in its graph at that point.
• Calculating Rates of Change: By understanding limits, we can define derivatives which represent instantaneous rates of changes.

The notation \( \lim_{{x \to c}} f(x) = L \) is commonly used to denote a limit. Understanding limits helps in visualizing how functions behave and interact, laying the groundwork for more complex analysis in calculus.

Formal Definition of a Limit

The formal definition of a limit, often called the \( \epsilon-\delta \) definition, provides a precise way to talk about the behavior of functions near a point. This definition is crucial for proving that a limit exists.To say that \( \lim_{{x \to c}} f(x) = L \), it means that for any small positive number \( \epsilon \), there is another small positive number \( \delta \) such that whenever \( x \) is within \( \delta \) units of \( c \) (but not equal to \( c \)), \( f(x) \) will be within \( \epsilon \) units of \( L \).

• Epsilon (\( \epsilon \)): Represents the margin of error or how close \( f(x) \) needs to be to \( L \).
• Delta (\( \delta \)): Indicates how close \( x \) should be to \( c \) to ensure \( f(x) \) stays within \( \epsilon \) of \( L \).

Understanding the \( \epsilon-\delta \) definition allows students to rigorously prove limits and analyze the precise behavior of functions as they approach specific points.

Proofs in Calculus

Proofs in calculus, such as proving limits using the formal definition, help solidify the understanding of mathematical concepts by demonstrating their truth thoroughly and rigorously. In the exercise provided, we have a proof using the formal definition of limits.Let's examine the key steps in proving \( \lim _{x \rightarrow c}(m x+b)=m c+b \):

• First, recognize the function and the limit point being considered. In the exercise, \( f(x) = mx + b \) and the limit supposed to be shown is \( mc + b \).
• Next, set up the inequality \(|(mx+b) - (mc+b)| < \epsilon\) to establish the \( \epsilon-\delta \) condition.
• Simplify this inequality to the form \(|m||x-c| < \epsilon\), which allows us to connect the limit with \( \epsilon \) and \( \delta \).
• Select \( \delta = \frac{\epsilon}{|m|} \) when \( m eq 0 \). This choice ensures that positioning \( x \) within \( \delta \) of \( c \) results in \( f(x) \) being within \( \epsilon \) of \( L \).

Through such structured methodologies, calculus proofs not only validate mathematical claims but also help reinforce logical reasoning and problem solving skills.