Limit Function Analysis: True Or False Statements

Hey guys! Today, we're diving into an exciting problem involving a function defined with an absolute value and exploring its limits. We'll break down the function, analyze its behavior as x approaches certain values, and determine which of the given statements are correct. Let's get started!

Understanding the Function

The function we're dealing with is:

$$f(x) = \frac{|9 - 3x|}{x - 3}$$

This function has an absolute value in the numerator, which means we need to consider different cases depending on the sign of the expression inside the absolute value, $9 - 3x$. Also, notice that the denominator is $x - 3$, so the function is not defined at $x = 3$. This is an essential point for our analysis. Understanding the nuances of such functions is key, so let's move forward step by step.

Analyzing the Statements

Now, let's evaluate each of the provided statements to determine their truthfulness.

Statement a: $\lim_{x \to 3^-} f(x)=3$

This statement deals with the left-hand limit as x approaches 3. When x approaches 3 from the left (i.e., $x < 3$), we have $9 - 3x > 0$. Therefore, $|9 - 3x| = 9 - 3x$. So, the function becomes:

$$f(x) = \frac{9 - 3x}{x - 3} = \frac{-3(x - 3)}{x - 3}$$

For $x \neq 3$, we can simplify this to:

$$f(x) = -3$$

Thus, $\lim_{x \to 3^-} f(x) = -3$. Therefore, statement a is false. Remember, it's crucial to correctly evaluate the sign inside the absolute value as it drastically changes the function's behavior.

Statement b: The value of $f(3)$ does not exist

As we noted earlier, the denominator of the function is $x - 3$. If we try to evaluate $f(3)$, we get:

$$f(3) = \frac{|9 - 3(3)|}{3 - 3} = \frac{0}{0}$$

Since division by zero is undefined, $f(3)$ does not exist. Therefore, statement b is true. Identifying points where the function is undefined is a critical skill in calculus.

Statement c: $\lim_{x \to 3^+} f(x)=3$

This statement deals with the right-hand limit as x approaches 3. When x approaches 3 from the right (i.e., $x > 3$), we have $9 - 3x < 0$. Therefore, $|9 - 3x| = -(9 - 3x) = 3x - 9$. So, the function becomes:

$$f(x) = \frac{3x - 9}{x - 3} = \frac{3(x - 3)}{x - 3}$$

For $x \neq 3$, we can simplify this to:

$$f(x) = 3$$

Thus, $\lim_{x \to 3^+} f(x) = 3$. Therefore, statement c is true. Analyzing limits from both sides is essential for a complete understanding of the function's behavior.

Statement d: $\lim_{x \to 3^-} f(x)=-3$

As we determined while analyzing statement a, when x approaches 3 from the left, $f(x) = -3$. Therefore, $\lim_{x \to 3^-} f(x) = -3$. So, statement d is true. Double-checking prior results can confirm the answers.

Conclusion

In summary, the correct statements are:

• b. The value of $f(3)$ does not exist
• c. $\lim_{x \to 3^+} f(x)=3$
• d. $\lim_{x \to 3^-} f(x)=-3$

Understanding limits involving absolute values requires careful consideration of the sign of the expression inside the absolute value. By analyzing the function from both the left and the right, and by identifying points where the function is undefined, we can accurately determine the behavior of the function and evaluate the given statements. Practice and attention to detail are your best friends in these kinds of problems.

Deeper Dive: Why Limits Matter

Okay, guys, so we've crunched the numbers and figured out which statements are true about the limit of this particular function. But you might be thinking, "Why do we even care about limits? What's the big deal?" That's a fair question, and it's important to understand the why behind the math. Limits are absolutely fundamental to calculus, and they underpin a whole host of concepts you'll encounter later on.

Think of it this way: limits allow us to investigate the behavior of a function near a point, even if the function itself isn't defined at that point. In our problem, $f(3)$ didn't exist, but we could still talk about what happened to the function as x got closer and closer to 3 from either side. This is super useful! It's like trying to understand what's on the edge of a cliff without actually falling off.

Here are a few key reasons why limits are so important:

• Defining Continuity: A function is continuous at a point if the limit of the function as x approaches that point exists, is finite, and is equal to the value of the function at that point. In simpler terms, you can draw the graph of a continuous function without lifting your pen. Limits are the foundation for understanding what it means for a function to be continuous, which is crucial in many applications. Continuity allows for predictability in mathematical models.
• Finding Derivatives: The derivative of a function at a point represents the instantaneous rate of change of the function at that point. The derivative is defined as a limit. Specifically, it's the limit of the difference quotient as the change in x approaches zero. So, without limits, we couldn't even define the derivative! Derivatives are used everywhere, from physics (calculating velocity and acceleration) to economics (analyzing marginal cost and revenue). The concept of derivatives is a cornerstone of applied mathematics.
• Calculating Integrals: Integration is essentially the reverse process of differentiation. It's used to find the area under a curve, the volume of a solid, and much more. The definite integral is defined as the limit of a Riemann sum. Again, without limits, we couldn't define the integral! Integration helps in calculating accumulated quantities.
• Understanding Asymptotic Behavior: Limits help us understand what happens to a function as x approaches infinity (or negative infinity). This is particularly useful for analyzing the long-term behavior of models. For example, we can use limits to determine whether a population growth model will stabilize, explode, or die out over time. Predicting the future of a system requires understanding asymptotic behavior.

So, the next time you encounter a limit problem, remember that you're not just doing abstract math. You're building a foundation for understanding some of the most powerful tools in mathematics and its applications. Don't underestimate the power of understanding limits.

Practical Tips for Solving Limit Problems

Alright, let's switch gears and talk about some practical tips and tricks that can help you tackle limit problems more effectively. These are the kinds of things that you'll pick up with practice, but it's helpful to have them in mind as you're working through problems.

• Always try direct substitution first: The first thing you should always do when evaluating a limit is to try plugging in the value that x is approaching. If you get a finite number, you're done! However, this often leads to indeterminate forms like 0/0 or ∞/∞, which means you need to do more work. Direct substitution is the quickest path to a solution, if it works.
• Factor and simplify: If direct substitution leads to an indeterminate form, try factoring the numerator and denominator and see if you can cancel out any common factors. This is a very common technique, especially when dealing with polynomials. In our original problem, we factored out a -3 (or a 3) to simplify the expression. Simplifying the expression can reveal the limit.
• Rationalize the numerator or denominator: If you have square roots in the expression, try multiplying the numerator and denominator by the conjugate of the expression containing the square root. This can help to eliminate the square roots and simplify the expression. Rationalization is a powerful tool for eliminating square roots.
• Use L'Hôpital's Rule: If you still have an indeterminate form after trying the above techniques, and if the limit has the form 0/0 or ∞/∞, you can apply L'Hôpital's Rule. This rule states that the limit of the quotient of two functions is equal to the limit of the quotient of their derivatives. Be careful, though: L'Hôpital's Rule only applies to indeterminate forms of the correct type. L'Hôpital's Rule can simplify complex indeterminate forms.
• Consider one-sided limits: If the limit as x approaches a point doesn't exist, it's often helpful to consider the one-sided limits (the limits as x approaches the point from the left and from the right). If the one-sided limits are different, then the overall limit doesn't exist. In our problem, we had to consider one-sided limits because of the absolute value. One-sided limits reveal different behaviors from each side.
• Think about the graph: Sometimes, it can be helpful to visualize the graph of the function. This can give you a better understanding of how the function is behaving as x approaches a certain value. There are many online graphing tools that you can use to plot functions. Visualizing the function can provide intuition.

Practice, practice, practice! The more limit problems you solve, the better you'll become at recognizing the different techniques and knowing when to apply them. Don't be afraid to make mistakes – that's how you learn! Consistent practice builds problem-solving skills.

So there you have it! A breakdown of the problem, an explanation of why limits are important, and some practical tips for solving limit problems. Keep practicing, and you'll be a limit master in no time!