I'm supposed to find the limit as x approaches infinity of (2-x-sinx)/(x+cosx). I've thought about using the squeeze theorem, but I'm not sure it applies here. Can someone point me in the right direction? Thanks.

Nevermind, I got it! I forgot that I could divide everything by the highest power x in the denominator.

I hope you got -1

To find the limit as x approaches infinity of a given rational function, such as (2-x-sinx)/(x+cosx), you're right that one approach is to use the squeeze theorem. However, before applying the squeeze theorem, we need to simplify the expression and analyze the behavior of its individual terms.

Let's start by simplifying the numerator and the denominator separately. We have:

Numerator: 2 - x - sin(x)
Denominator: x + cos(x)

Now, let's look at each term independently to understand their behavior as x approaches infinity.

1. Numerator:
As x approaches infinity, the term "2" doesn't change. The "-x" term approaches negative infinity, and the "sin(x)" term oscillates between -1 and 1 as x increases without bound.

2. Denominator:
As x approaches infinity, the "x" term grows without bound. The "cos(x)" term oscillates between -1 and 1 but does not affect the overall behavior of the function.

Now, let's consider the signs of the terms as x approaches infinity:

1. Numerator:
- The "2" term is positive.
- The "-x" term is negative.
- The "sin(x)" term oscillates between -1 and 1.

2. Denominator:
- The "x" term is positive.
- The "cos(x)" term oscillates between -1 and 1.

Next, let's consider the magnitudes of the terms as x approaches infinity. Here, we'll focus on the dominant term that will have the most significant effect on the overall behavior of the function.

1. Numerator:
- The "-x" term has a larger magnitude than the "2" term and the "sin(x)" term.

2. Denominator:
- The "x" term has a larger magnitude than the "cos(x)" term.

Now, we can use this analysis to apply the squeeze theorem. Since the numerator and denominator both have terms that approach infinity, we can't directly apply the squeeze theorem in this case.

To handle such situations, we can try dividing both the numerator and denominator by the highest power of x present in the function. In this case, it's x. Dividing both numerator and denominator by x, we have:

(2/x - 1 - sin(x)/x) / (1 + cos(x)/x)

Now, as x approaches infinity, the terms "2/x" and "sin(x)/x" both approach zero, and the term "cos(x)/x" also approaches zero. This simplifies the expression to:

(0 - 1 - 0) / (1 + 0) = -1

Therefore, the limit as x approaches infinity of (2-x-sinx)/(x+cosx) is -1.

So, to recap:

1. Simplify the expression by dividing both numerator and denominator by the highest power of x present in the function.
2. Analyze the behavior of each term as x approaches infinity, considering their signs and magnitudes.
3. Apply the squeeze theorem if possible, or simplify the expression further to evaluate the limit.

I hope this explanation helps! Let me know if you have any further questions.