Finding Limits

In Exercises ...

Question

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

g(x) = 1/(2 + (1/x))

Accepted Answer

Welcome back, everyone. Calculate the limit of H of X equals 1 divided by 15 + 3 divided by XS X approaches positive or negative infinity. So let's write down the limit. As X approaches infinity, that's the first part, right? We're going to first of all solve as X approach as positive infinity, and we have a fraction, 1 divided by 15 plus 3 divided by X. We can apply the properties of limits and distribute the limit for each part of the fraction. So first of all, we get limited SX approaches infinity of 1. Divided by Limit as X approaches infinity of 15. Plus limit. As x approaches infinity of 3 divided by X. Now, what can we know? Well, the top limit is the limit of a constant which is independent of X. So the answer is that constant, which is 1, and for the denominator, we have limit as X approaches infinity of 15, so another constant which gives us 15. And now the the final limit which contains X is we divided by X. Let's recall that the limit as x approaches positive or negative infinity of a number a divided by X. As long as A is not equal to 0. Is going to give us 0. In other words, a non-zero number divided by an infinitely large number leads to the limit of 0, whether we're approaching positive or negative infinity. This is exactly what we have for our limit. S X approaches infinity of 3 divided by X. It is going to be equal to 0, right, because we're dividing 3 by an infinitely large positive number. So we get 1 divided by 15 + 0, which gives us 1 divided by 15, and that's how we first answer. Now for the second limit, we get limit as X approaches negative infinity. Of H of X, the same function, and we can basically apply the same properties. So this gives us a limit as X approaches negative infinity of 1. Divided by limit as X approaches negative infinity of 15. Plus Limet As X approaches negative infinity of 3 divided by X. For each constant we already know that they are independent of X, so it doesn't matter if we are approaching a negative infinity, positive infinity, or any other X value, we're going to get those constants as 1 in the numerator, 15 in the denominator, and for the final limit, the only difference is that we're approaching negative infinity, but we divided by negative infinity as the limit is also going to be equal to 0. Right? Based on the definition. It doesn't matter if we are dividing by an infinitely large positive or negative number, the limit is going to be equal to 0. So we get the same answer, that's going to be 1 divided by 15. So we can conclude that the limit as X approaches positive or negative infinity of H of X is going to be equal to 1 divided by 15, which is our final answer to this problem. Thank you for watching.

Finding Limits

In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)

g(x) = 1/(2 + (1/x))

Key Concepts

Limits

Behavior at Infinity

Rational Functions

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