Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ h(x)=\frac{3}{x-3} $$

Step-by-Step Solution

Answer

The function is not continuous at 3 because it is undefined and the limit does not exist at that point.

Step 1: Check the definition of continuity

A function \( f(x) \) is continuous at a point \( a \) if the following three conditions are met: 1. \( f(a) \) is defined.2. \( \lim_{{x \to a}} f(x) \) exists.3. \( \lim_{{x \to a}} f(x) = f(a) \).

Step 2: Determine if \( h(x) \) is defined at 3

Plug \( x = 3 \) into the function \( h(x) = \frac{3}{x-3} \):\[ h(3) = \frac{3}{3-3} = \frac{3}{0} \]Since dividing by zero is undefined, \( h(3) \) is not defined.

Step 3: Evaluate the limit as \( x \to 3 \) for \( h(x) \)

Find the limit of \( h(x) \) as \( x \) approaches 3:\[ \lim_{{x \to 3}} \frac{3}{x-3} \]As \( x \) approaches 3, the denominator \( x-3 \) approaches 0, leading the expression to approach infinity. Therefore, the limit does not exist.

Step 4: Conclude Continuity at \( x = 3 \)

Based on Step 2, \( h(3) \) is not defined, and from Step 3, the limit as \( x \to 3 \) does not exist. Thus, the function \( h(x) \) is not continuous at \( x = 3 \) because it fails both the conditions of the definition of continuity.

Key Concepts

Understanding Limits

In calculus, limits help us understand the behavior of a function as the input approaches a particular value. Essentially, the limit evaluates what the output value of a function becomes as we get near a specific point, even if the function itself isn't defined exactly at that point.

For the function \[ h(x) = \frac{3}{x-3} \]as we approach \( x = 3 \), the expression approaches infinity. This is because the denominator \((x-3)\) becomes very close to zero, which means the fraction itself becomes very large or very small (depending on the direction of approach).

It is important to know:

• If a limit approaches a finite number as \( x \to a \), it means the function can potentially be continuous, but other conditions must also be met.
• If a limit does not exist (i.e., approaches infinity or varies wildly), the function cannot be continuous at that point.

Understanding limits is crucial for studying function behavior, especially in cases where the function isn't well-behaved or defined at particular points.

Evaluating Function Values

Function evaluation is the process of calculating the output for a given input in the function, like plugging specific values into the equation.

For the function \[ h(x) = \frac{3}{x-3} \]at \( x = 3 \), the expression \( h(3) \) turns into \( \frac{3}{0} \), indicating division by zero, which is mathematically undefined.

Steps for function evaluation:

• Substitute the input value directly into the function where \( x \) appears.
• Simplify the expression and assess if it results in a valid numeric answer.
• If it results in division by zero or any other undefined operation, the function at that particular point remains undefined.

Being able to evaluate function values helps determine if they are defined or if there are any points where computations fail.

Handling Undefined Values

Undefined values arise in mathematics when calculations fall outside standard operations, such as dividing by zero.

In \[ h(x) = \frac{3}{x-3} \]at \( x = 3 \), the expression becomes \( \frac{3}{0} \), a classic case of undefined behavior due to division by zero.

Key Points on Undefined Values:

• Function outputs can be undefined at specific points, often involving operations like division by zero, taking the logarithm of a negative number, etc.
• When a value is undefined, it indicates a discontinuity in the function's graph at that point.
• Recognizing undefined values is crucial in determining a function's continuity and where it might break down or become asymptotic.

Dealing with undefined values helps in plotting accurate graphs and understanding where a function might need special handling or notation.