Finding Limits And Oblique Asymptotes For F(x)
Hey guys! Let's dive into a cool math problem where we're given a function and need to figure out a couple of things about it. Specifically, we're going to find a limit and an oblique asymptote. Don't worry if these terms sound intimidating; we'll break it down step by step so it's super clear.
Understanding the Problem
Okay, so we've got this function: f(x) = (x^2 - x - 3) / (x - 3). Our mission, should we choose to accept it (and we do!), is twofold:
- Figure out what happens to f(x) as x approaches 3 from the left side. This is what the limit question is all about. We write it mathematically as lim (x→3-) f(x) = ?. The little minus sign (-) above the 3 is crucial – it tells us we're only looking at values of x that are slightly less than 3.
- Find the oblique asymptote of f(x). An oblique asymptote is like a guideline that the function follows when x gets really, really big (or really, really small). It's a straight line that the function gets closer and closer to, but never actually touches. Think of it like a long-term trend for the function.
So, buckle up, grab your calculators (or your mental math muscles!), and let's get started!
Calculating the Limit: lim (x→3-) f(x)
Let's tackle the first part: finding the limit as x approaches 3 from the left. This is where things can get a little tricky, but don't sweat it – we'll navigate it together.
Why Can't We Just Plug in x = 3?
The first thing you might think to do is simply plug in x = 3 into our function. Makes sense, right? But if we try that, we get:
f(3) = (3^2 - 3 - 3) / (3 - 3) = (9 - 3 - 3) / 0 = 3 / 0
Uh oh! We've got a problem. Dividing by zero is a big no-no in the math world. It's undefined, which means we can't directly evaluate the function at x = 3. This tells us that there's something interesting happening at x = 3, and we need to investigate further.
Approaching from the Left
Since we can't plug in x = 3 directly, we need to think about what happens as x gets really close to 3, but stays less than 3. Let's try plugging in some values that are close to 3 from the left, like 2.9, 2.99, and 2.999, and see what happens to f(x):
- If x = 2.9, then f(2.9) = (2.9^2 - 2.9 - 3) / (2.9 - 3) = (8.41 - 2.9 - 3) / (-0.1) = 2.51 / (-0.1) = -25.1
- If x = 2.99, then f(2.99) = (2.99^2 - 2.99 - 3) / (2.99 - 3) = (8.9401 - 2.99 - 3) / (-0.01) = 2.9501 / (-0.01) = -295.01
- If x = 2.999, then f(2.999) = (2.999^2 - 2.999 - 3) / (2.999 - 3) = (8.994001 - 2.999 - 3) / (-0.001) = 2.995001 / (-0.001) = -2995.001
Do you see a pattern here? As x gets closer and closer to 3 from the left, f(x) is getting more and more negative. It's diving down towards negative infinity! We can write this mathematically as:
lim (x→3-) f(x) = -∞
So, the answer to the first part of our problem is negative infinity. Cool, right?
Finding the Oblique Asymptote
Now, let's move on to the second part: finding the oblique asymptote of f(x). This might sound like a complicated process, but it's actually pretty straightforward once you know the trick.
What is an Oblique Asymptote, Exactly?
Before we jump into the calculations, let's make sure we're all on the same page about what an oblique asymptote is. As we mentioned earlier, it's a slanted line that a function approaches as x gets very large (positive or negative). It only exists when the degree of the numerator of a rational function is exactly one more than the degree of the denominator.
In our case, f(x) = (x^2 - x - 3) / (x - 3). The numerator (x^2 - x - 3) has a degree of 2 (because the highest power of x is 2), and the denominator (x - 3) has a degree of 1. Since 2 is exactly one more than 1, we know that f(x) does have an oblique asymptote.
The Magic of Polynomial Long Division
The key to finding the oblique asymptote is polynomial long division. Remember that from algebra? We're going to divide the numerator (x^2 - x - 3) by the denominator (x - 3).
Here's how it works:
x + 2
x - 3 | x^2 - x - 3
-(x^2 - 3x)
-------------
2x - 3
-(2x - 6)
---------
3
Okay, let's break down what just happened:
- We set up the long division problem with x^2 - x - 3 inside the division symbol and x - 3 outside.
- We asked ourselves, "What do we need to multiply x by to get x^2?" The answer is x, so we wrote x above the -x term.
- We multiplied x by (x - 3) to get x^2 - 3x and wrote that below x^2 - x. We then subtracted (x^2 - 3x) from (x^2 - x), which gave us 2x. We brought down the -3.
- We asked ourselves, "What do we need to multiply x by to get 2x?" The answer is 2, so we wrote + 2 next to the x above.
- We multiplied 2 by (x - 3) to get 2x - 6 and wrote that below 2x - 3. We then subtracted (2x - 6) from (2x - 3), which gave us a remainder of 3.
The Oblique Asymptote Revealed!
So, what does all this mean? Well, the result of our long division tells us that:
(x^2 - x - 3) / (x - 3) = x + 2 + 3 / (x - 3)
This is the crucial step! The oblique asymptote is the quotient we got from the long division, ignoring the remainder. In this case, the quotient is x + 2. Therefore, the oblique asymptote of f(x) is the line:
y = x + 2
That's it! We've found the oblique asymptote. Awesome!
Wrapping It Up
Alright, guys, we did it! We successfully tackled a limit problem and found an oblique asymptote. Let's recap what we've learned:
- We found that lim (x→3-) f(x) = -∞, meaning that as x approaches 3 from the left, the function dives down to negative infinity.
- We used polynomial long division to rewrite our function and identify its oblique asymptote, which is the line y = x + 2.
These are important concepts in calculus and precalculus, so give yourself a pat on the back for working through this! If you have any questions, don't hesitate to ask. Keep practicing, and you'll be a math whiz in no time!Understanding the Problem
Calculating the Limit: lim (x→3-) f(x)
Why Can't We Just Plug in x = 3?
Approaching from the Left
Finding the Oblique Asymptote
What is an Oblique Asymptote, Exactly?
The Magic of Polynomial Long Division
The Oblique Asymptote Revealed!
Wrapping It Up
