a. Analyze

Question

a. Analyze ${\displaystyle\lim_{x o\infty}{f(x)}}$ and ${\displaystyle\lim_{x o-\infty}{f(x)}}$ for each function.

$f (x) = \frac{1 + x - 2 x^{2} - x^{3}}{x^{2} + 1}$

Accepted Answer

Welcome back, everyone. Find the following limits for the function F of X equals 4 + x minus 3x2 + x to the power of 4 divided by 2 + X2, and we want to find two limits limit of F of X X approaches positive and negative infinity. So let's begin with the first one. It says limit as x approaches positive infinity, and we're going to substitute the function. It's 4 + x minus 3x2 + x to the power of 4th. And now in the denominator we have 2 + X2. Now if we substitute infinity and if we simply apply the right substitution. Well, we're going to get 4 plus infinity minus infinity plus infinity divided by infinity. So we essentially have. And indeterminous form, infinity minus infinity divided by infinity. Because this is an indeterminate form and we have a rational function consisting of two polynomials, one of the rules is to divide both sides of the given fraction by the variable with the highest exponent in the denominator, and that's x2d, right? So what we're going to do is basically divide both sides by X2, and this gives us. Limit as X approaches infinity of 4 divided by X2 plus x divided by X2 minus 3 x 2 divided by X2. Plus x 4th divided by X2 and in the denominator, we end up with 2 divided by X2 + X2 divided by X2. And now let's simplify. The limit becomes limit as x approaches infinity of 4 divided by X2 + 1 divided by X minus 3. Plus x squared. For the denominator we get 2 divided by X2 + 1. And now let's apply the right substitution once again. As X approaches infinity, a number divided by X2 will approach 0, right, because the denominator grows. Towards infinity and the numerator doesn't change. So we're going to cross out all of the terms that are fractional terms, a number divided by X raises to some exponent. All of those will approach 0. And from here as we can see we end up with. -3 plus X2, which is infinity squared or basically infinity, divided by 1. And of course this gives us positive infinity. Now when we analyze the limit as x approaches negative infinity, nothing really changes. All of those terms will approach 0 but from different sides, right? So essentially we will have the same limit of F of X. Which is going to be equal to infinity. Notice that we will perform the same calculations, the same simplification of the given a rational function. And then each term will still approach 0. Now, for example, 1 divided by X will approach 0 from the left, but it will have no effect on the calculation, and we will still get infinity. So the answer to this problem is infinity for each limit. Limit as x approaches positive infinity is infinity and limit as x approaches negative infinity is also positive infinity. Thank you for watching.

a. Analyze ${\displaystyle\lim_{x\to\infty}{f(x)}}$ and ${\displaystyle\lim_{x\to-\infty}{f(x)}}$ for each function.

$f (x) = \frac{1 + x - 2 x^{2} - x^{3}}{x^{2} + 1}$

Key Concepts

Limits at Infinity

Rational Functions

Dominant Terms

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