Complete the following steps for the given functions. b. Find the vertical asymptotes of f f f (if any). f ( x ) = 4 x 3 + 4 x 2 + 7 x + 4 x 2 + 1 f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1} f ( x ) = x 2 + 1 4 x 3 + 4 x 2 + 7 x + 4

Welcome back everyone. In this problem, we want to determine the vertical asymptotes. If any of the function H of X equals two X cube plus nine X squared minus five, divided by three X squared plus 27 A says X equals negative nine B says X equals nine C says X equals three and negative three. And D says the function has no vertical Asymptote. Now, what do we know about vertical asymptotes? Well, they occur where the function approaches infinity or negative infinity as the input approaches a certain value, these points often correspond to the denominator of a rational function being zero provided that the numerator does not also become zero at the same point. So if we're going to find a vertical Asymptote, first, we need to figure out the value that makes our denominator zero, then we need to check if it also makes the numerator zero. And if it does not, we can verify the behavior of our function from the left and the right hand side. So first let's try to find the value of our denominator or sorry, the value that will make the denominator of our function zero. OK. Now if we let three X squared plus 27 equal zero, OK, then that means three X squared equals negative 27. If we divide both sides by three X squared equals negative nine and X is going to be equal to the square root of negative nine. But we've run into a problem here. We can't find the square root of a negative number. Or in other words, the square root of a negative number is not real. OK. So that means there are no real solutions. OK. To this equation, since there are no real solutions, that means there is no real value of X that will make the denominator zero. And if that's the case, that means the given function has no vertical Asymptote because remember that the point near far or vertical Asymptote often corresponds to the denominator being zero. Thus D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

1. Limits and Continuity

Introduction to Limits

Calculus

1. Limits and Continuity

Introduction to Limits

2:38 minutes

Problem 2.5.55b

Textbook Question

Complete the following steps for the given functions.

b. Find the vertical asymptotes of $f$ (if any).

$f\left(x\right)=\frac{4x^3+4x^2+7x+4}{x^2+1}$

Verified step by step guidance

Identify the function given: $ f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} $. This is a rational function, which is a ratio of two polynomials.

To find vertical asymptotes, we need to determine where the denominator is equal to zero, as these are the points where the function is undefined.

Set the denominator equal to zero: $ x^2 + 1 = 0 $.

Solve the equation $ x^2 + 1 = 0 $ for $ x $. Notice that $ x^2 = -1 $, which has no real solutions since the square of a real number cannot be negative.

Conclude that there are no vertical asymptotes for the function $ f(x) = \frac{4x^3 + 4x^2 + 7x + 4}{x^2 + 1} $ because the denominator does not have any real roots.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertical Asymptotes

Vertical asymptotes occur in rational functions where the denominator approaches zero while the numerator does not. These points indicate where the function's value tends to infinity or negative infinity, leading to a discontinuity. To find vertical asymptotes, set the denominator equal to zero and solve for the variable, ensuring that the numerator does not also equal zero at those points.

Rational Functions

A rational function is a function represented by the ratio of two polynomials. The general form is f(x) = P(x)/Q(x), where P and Q are polynomials. Understanding the behavior of rational functions is crucial for analyzing their asymptotic behavior, including identifying vertical and horizontal asymptotes, as well as points of discontinuity.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. They play a significant role in calculus, particularly in determining the behavior of rational functions. The degree of the polynomial affects the function's end behavior and can influence the presence of asymptotes.

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