Solving Limits At Infinity: Step-by-Step Guide & Examples

Hey guys! Ever find yourselves staring blankly at limit problems that stretch off to infinity? Don't worry, you're not alone! Limits at infinity might seem intimidating, but with a few tricks and a bit of practice, you can totally nail them. In this guide, we'll break down how to tackle these problems, walking through two examples step-by-step. So, let's dive in and conquer those infinite limits!

Understanding Limits at Infinity

Before we jump into the problems, let's quickly recap what a limit at infinity actually means. When we say $\lim_{x \to \infty} f(x)$, we're asking: "What value does the function f(x) approach as x gets incredibly large?" It's like watching a car speed down a never-ending road – where is it heading? Understanding this concept is crucial for tackling the problems we're about to solve.

Now, the key idea when dealing with rational functions (that is, fractions where the numerator and denominator are polynomials) at infinity is to focus on the highest powers of the variable. These are the terms that will dominate the function's behavior as x gets huge. Think of it like this: if you're comparing the heights of an ant and an elephant, the elephant's height is going to be much more significant! Similarly, the highest power terms in a polynomial will overshadow the lower power terms as x approaches infinity. Keep this in mind, and you'll be well on your way to mastering limits at infinity.

Problem 1: Finding the Limit of a Rational Function

Let's tackle our first problem: $\lim_{t \to \infty} \frac{4t^5 +7t^2 +t}{2t^4 + 5t^3 +1}$.

Step 1: Identify the Highest Powers

First, we need to pinpoint the highest power of t in both the numerator and the denominator. Looking at the expression, we see that the highest power in the numerator is t⁵, and the highest power in the denominator is t⁴. These are our "elephants" – the terms that will dictate the limit's behavior.

Step 2: Divide by the Highest Power in the Denominator

This is the crucial step. We're going to divide both the numerator and the denominator by t⁴. Why t⁴? Because it's the highest power in the denominator. This might seem like a random move, but it's a clever trick to simplify the expression and make the limit easier to evaluate. Doing this, we get:

$\lim_{t \to \infty} \frac{(4t^5 +7t^2 +t) / t^4}{(2t^4 + 5t^3 +1) / t^4}$

Now, let's distribute the division:

$\lim_{t \to \infty} \frac{4t^5/t^4 + 7t^2/t^4 + t/t^4}{2t^4/t^4 + 5t^3/t^4 + 1/t^4}$

Step 3: Simplify

Time to simplify those fractions by canceling out the powers of t:

$\lim_{t \to \infty} \frac{4t + 7/t^2 + 1/t^3}{2 + 5/t + 1/t^4}$

Notice what's happening here! We've transformed the expression into something much more manageable. We have terms like 7/t², 1/t³, 5/t, and 1/t⁴. As t approaches infinity, what happens to these terms? They all approach zero! This is the key to solving limits at infinity.

Step 4: Evaluate the Limit

Now, let's consider what happens as t approaches infinity. The terms 7/t², 1/t³, 5/t, and 1/t⁴ all go to zero. So, our expression becomes:

$\lim_{t \to \infty} \frac{4t + 0 + 0}{2 + 0 + 0} = \lim_{t \to \infty} \frac{4t}{2} = \lim_{t \to \infty} 2t$

As t gets infinitely large, 2t also gets infinitely large. Therefore, the limit is:

$\lim_{t \to \infty} 2t = \infty$

So, the answer to our first problem is infinity! See? Not so scary after all. By focusing on the highest powers and dividing, we simplified the expression and easily found the limit.

Problem 2: Tackling a More Complex Rational Function

Alright, let's level up! Our second problem is: $\lim_{x \to \infty} \frac{(1-2x)^3}{(x-1)(2x^2+x+1)}$. This one looks a bit more intimidating, but don't sweat it. We'll use the same principles we learned in the first problem, with a few extra steps.

Step 1: Expand the Expressions

Before we can identify the highest powers, we need to expand the numerator and denominator. Let's start with the numerator, (1-2x)³. Remember the binomial expansion or the formula (a-b)³ = a³ - 3a²b + 3ab² - b³. Applying this, we get:

(1-2x)^3 = 1 - 6x + 12x^2 - 8x^3

Now, let's expand the denominator, (x-1)(2x²+x+1). We can do this by carefully multiplying each term:

(x-1)(2x^2+x+1) = 2x^3 + x^2 + x - 2x^2 - x - 1 = 2x^3 - x^2 - 1

So, our limit now looks like this:

$\lim_{x \to \infty} \frac{1 - 6x + 12x^2 - 8x^3}{2x^3 - x^2 - 1}$

Step 2: Identify the Highest Powers

Now that we've expanded the expressions, we can easily identify the highest power of x in both the numerator and the denominator. In this case, it's x³ in both the numerator and the denominator.

Step 3: Divide by the Highest Power in the Denominator

Just like before, we'll divide both the numerator and the denominator by the highest power in the denominator, which is x³:

$\lim_{x \to \infty} \frac{(1 - 6x + 12x^2 - 8x^3) / x^3}{(2x^3 - x^2 - 1) / x^3}$

Distribute the division:

$\lim_{x \to \infty} \frac{1/x^3 - 6x/x^3 + 12x^2/x^3 - 8x^3/x^3}{2x^3/x^3 - x^2/x^3 - 1/x^3}$

Step 4: Simplify

Time to simplify those fractions:

$\lim_{x \to \infty} \frac{1/x^3 - 6/x^2 + 12/x - 8}{2 - 1/x - 1/x^3}$

Again, notice the pattern! We have terms like 1/x³, 6/x², 12/x, and 1/x³. As x approaches infinity, these terms will all go to zero.

Step 5: Evaluate the Limit

As x approaches infinity, the terms 1/x³, 6/x², 12/x, and 1/x³ all approach zero. Our expression simplifies to:

$\lim_{x \to \infty} \frac{0 - 0 + 0 - 8}{2 - 0 - 0} = \lim_{x \to \infty} \frac{-8}{2} = -4$

So, the answer to our second problem is -4! We tackled a more complex rational function by first expanding the expressions, then using the same technique of dividing by the highest power and simplifying. You've got this!

Key Takeaways for Conquering Limits at Infinity

Okay, guys, let's recap the key steps to solve limits at infinity involving rational functions:

1. Identify the Highest Powers: Pinpoint the highest power of the variable in both the numerator and the denominator. These are the terms that will dominate the function's behavior as the variable approaches infinity.
2. Divide by the Highest Power in the Denominator: This is the crucial step that simplifies the expression and allows you to evaluate the limit.
3. Simplify: Cancel out the powers of the variable and rewrite the expression.
4. Evaluate the Limit: As the variable approaches infinity, terms with the variable in the denominator will approach zero. Use this to simplify the expression and find the limit.

Remember this crucial concept: When dealing with limits at infinity of rational functions:

• If the highest power in the numerator is greater than the highest power in the denominator, the limit is either infinity or negative infinity (depending on the leading coefficients).
• If the highest power in the denominator is greater than the highest power in the numerator, the limit is zero.
• If the highest powers in the numerator and denominator are the same, the limit is the ratio of the leading coefficients.

Practice Makes Perfect!

Limits at infinity might have seemed tricky at first, but with these steps and some practice, you'll be solving them like a pro. The most important thing is to understand the underlying concepts and apply the techniques consistently. So, grab some more problems, work through them step-by-step, and build your confidence. You've got this, guys! Keep up the awesome work, and you'll be mastering calculus in no time!