Problem 2 - Calculus for Biology and Medicine

Question

In Problems 1-4, show that each function is continuous at the given value. $$ f(x)=-x, c=1 $$

Step-by-Step Solution

Answer

The function \( f(x) = -x \) is continuous at \( x = 1 \) because \( \lim_{x \to 1} f(x) = f(1) = -1 \).

Step 1: Define Continuity at a Point

To show that a function is continuous at a given point \( c \), the limit of the function as \( x \) approaches \( c \) must equal the function's value at \( c \). Mathematically, this means \( \lim_{x \to c} f(x) = f(c) \).

Step 2: Substitute the Given Function and Point

For the function \( f(x) = -x \) and \( c = 1 \), we first calculate the value of the function at \( c \): \( f(1) = -1 \).

Step 3: Evaluate the Limit of the Function as x Approaches c

Calculate \( \lim_{x \to 1} f(x) = \lim_{x \to 1} (-x) \). Since \( f(x) = -x \) is a linear function, its limit as \( x \to 1 \) is simply \(-1\).

Step 4: Compare the Limit with the Function Value

Compare the limit \( \lim_{x \to 1} f(x) = -1 \) with the function value \( f(1) = -1 \). Both the limit and the function value are equal.

Key Concepts

Understanding Limits

In calculus, the concept of limits plays a crucial role in determining the behavior of functions as they approach a particular point. The limit essentially provides a way to predict where the function is heading as the input, often denoted as \( x \), gets closer and closer to a specific value. When we speak of the limit of a function \( f(x) \) as \( x \to c \), we are asking what value \( f(x) \) is approaching when \( x \) gets increasingly close to \( c \).
There are a few central characteristics to remember about limits:

• Limits examine the trend of a function as inputs approach a given value, without necessarily reaching it.
• They help us understand what happens with function values very close to a certain point.
• Finding limits is often the first step when checking for continuity.

In our example, we evaluated \( \lim_{x \to 1} (-x) \), and since this limit equals \(-1\), it matches the function's value at \( x = 1 \). This concurrence is essential for proving continuity.

Function Evaluation

Function evaluation refers to substituting a specific value into a function and computing the result, which is crucial for analyzing the function's characteristics at particular points.
Let's go through the process of evaluating the function \( f(x) = -x \) at \( x = 1 \). By substituting into the function, we perform the operation \( f(1) = -(1) \), which gives us \(-1\).

• It’s a straightforward process but forms the backbone of many higher-level calculus concepts.
• This evaluated result of \( f(1) = -1 \) allows us to compare it with the limit we previously calculated.
• The accuracy of function evaluation is crucial for determining continuity.

In our exercise, function evaluation showed that at \( x = 1 \), \( f(1) \) gives exactly the same output as the limit we found, which indicates that there are no jumps or breaks in the graph at that point.

Point of Continuity

A point of continuity is where a function is seamless and unbroken; the graph is smooth at that particular point. For a function to be continuous at a point \( c \), three conditions must be satisfied:

• The function \( f(x) \) must be defined at \( c \). This means \( f(c) \) exists.
• The limit of \( f(x) \) as \( x \cc \to c \) should exist.
• The limit of \( f(x) \) as \( x \to c \) must equal the function value \( f(c) \).

In simpler terms, continuity ensures that there are no gaps or jumps in the function's graph at \( x = c \).
In our solution, all three conditions were met for \( f(x) = -x \) at \( c = 1 \):

• We calculated \( f(1) = -1 \), so the function is defined at \( x = 1 \).
• We found \( \lim_{x \to 1} (-x) \) which also resulted in \(-1\), confirming the limit exists at the point.
• The limit and function value both agree, showing no disruptions.

Therefore, \( f(x) = -x \) is continuous at \( x = 1 \), a valuable insight into the behavior of linear functions.