Limits of Rational Functions

Question

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (2x + 3)/(5x + 7)

Accepted Answer

Welcome back, everyone. In this problem, we want to calculate the limit of the function P X equals 4 X + 11 divided by 3 X + 8 as X approaches infinity and as X approaches negative infinity. A says both answers are negative 4/3. B says as it approaches infinity, it's 4/3, while as it approaches negative infinity, it is 4/3. C says it's negative 4/3 and 4/3 respectively, and D says both are 4/3. Now, before we calculate the limit, let's factor out X from PF X, OK? So we know that PF X equals 4 X + 11 divided by 3 X + 8. When we factor with X, we'll get X multiplied by 4 + 11 divided by X in our numerator. And in our denominator, we'll get X multiplied by 3 + 8 divided by X. And now when we factor out X, then we get PF X to be 4 + 11 divided by X, all divided by 3 + 8 divided by X. Know that we have this value for PF X, then let's go ahead and try to find our limit, OK? And now we've done that because here we we've been able to cancel out X where X is not equal to 0, OK. Now, as X, let's start with our first case, as X approaches infinity, that means we're going to be finding the limit as X approaches infinity of 4 + 11 divided by X divided by 3 + 8 divided by X. Now if we think about it here as X gets bigger and it is the denominator of our quotient, that means the value will 10 to 0, OK? Because as the denominator gets larger, then our number gets smaller, OK? So that means then that this limit is going to be equal to 4 + 0 divided by 3 + 0. OK, which is equal to 4/3. Now how about as X approaches negative infinity. Well, we can imagine the same thing is going to happen, OK? So now we know that our limit as X approaches negative infinity of PF X, OK? Our number again is also going to 10 to 0, OK? That is our number where our denominator is X. So in this case, this will also work out to 4 + 0 divided by 3 + 0, which equals 4/3. Therefore, the limit of PF X as X approaches infinity and the negative infinity is going to be 4/3. If we look at our answer choices, that means D is the correct answer. Thanks for watching everyone. I hope this video helped.

Limits of Rational Functions

In Exercises 13–22, find the limit of each rational function (a) as x → ∞ and (b) as x → −∞. Write ∞ or −∞ where appropriate.

f(x) = (2x + 3)/(5x + 7)

Key Concepts

Limits

Rational Functions

End Behavior

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