Find the vertical asymptotes. For each vertical asymptote x=a,...

Question

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).

f(x) = (2x³ + 10x² + 12x) / (x³ + 2x²)

Accepted Answer

Welcome back, everyone. Identify any vertical asymptotes of the function F of X is equal to 5 x cubed plus 30X2 + 40 X divided by X cubed plus 4 x 2. Evaluate limit of FX as X approaches A from the left and A from the right for each vertical asymptote X equals a. So first of all, what we're going to do is analyze the given function. It is a rational function. It has a numerator of 5 x cubed. Plus 30 x squared plus 40 X. And in the denominator we have x cubed plus 4 x 2d. Whenever we are considering vertical asymptotes, the first step is to make sure that we have completely factored out the numerator and the denominator. So first of all, what we can do is consider. The numerator and we can notice that we can factor out 5 x which gives us X2 + 6 x + 8 as the other factor and then in the denominator we can factor out X2d which gives us X + 4 as the remaining factor. And now in the numerator we're going to Actor out the quadratic polynomial, which can be factored as x + 4 multiplied by x + 2. Let's rewrite the denominator x2 multiplied by x + 4. And what we can notice is that we can first of all cancel out X +4, which gives us a whole, and then we can cancel out X and X2, right? So the simplified form is going to be 5. Multiplied by X + 2. Divided by X. Now This is where we begin solving the problem. A vertical asymptote essentially makes the denominator equal to 0. So in this case, X must be equal to 0, and we also want to make sure that the numerator is not equal to 0 at the same value of x. So 5 multiplied by X + 2 is not equal to 0 because 0 + 2 gives us 22 multiplied by 5 gives us 10, so X equals 0. Is indeed a vertical asymptote. And now our next step is to evaluate the two limits. The first one is limit as x approaches 0 from the left because A is equal to 0, right? A is equal to 0, that's the vertical asymptote. And we're going to use the simplified form 5 multiplied by X + 2. Divided by X. Now, if we use direct substitution in the numerator, we're going to get 5 multiplied by 0. Plus 2, and in the denominator, we're going to get 0 from the left. The reason why we didn't write 0 from the left in the numerator is simply because 0 + 2, whether it's from the left or from the right, gives us a limit of 2. So we end up with 5 multiplied by 2, which is 10. It is a positive number divided by 0 from the left. A positive number divided by a negative number gives us a negative number, and a number divided by 0 gives us a limit of infinity. So we are going to get negative infinity for the first limit. And now as X approaches 0 from the right. We're going to take 5 multiplied by X plus 2 divided by X. Let's use the direct substitution 5 multiplied by 0 + 2. Divided by 0 from the right. So in this case, our numerator is the same, it is equal to 10, but our denominator is 0 from the right, so we have a ratio of two positive numbers which now gives us positive infinity. And that's it. We have our answers. So the vertical asymptote has a form of X equals 0, and the two limits are equal to negative infinity as X approaches 0 from the left. From the right, the limit is equal to positive infinity. Thank you for watching.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = (2x³ + 10x² + 12x) / (x³ + 2x²)

Key Concepts

Vertical Asymptotes

Limits

Rational Functions

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