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.)

f(x) = 2/x − 3

Accepted Answer

Welcome back, everyone. Calculate the limit of H X equals 3 divided by X minus 4 S X approaches positive and negative infinity. Let's begin with the first limit. Limit SX approaches positive infinity of 3 divided by X minus 4. We can begin with the right substitution or basically we can apply the properties of limits, right? So what we're going to do is simply distribute the limit because we have a difference. So we get two limits, limit as X approaches infinity. Of 3 divided by X minus limit as X approaches infinity of 4. Let's consider the first limit. We have 3 divided by infinity. By definition, whenever we divide a number by an infinitely large number, we get 0. We just want to make sure that our numerator is not 0 and it is not 0. It is 3, right? So 3 divided by an infinitely large number leads to the limit of 0. And for the second limit, we have limit as X approaches infinity of 4. So we end up with 4 because 4 is a constant. It is independent of X. So, 0 minus 4. Gives us -4. That said, we have our first limit, and now we can continue with the second one. Limit as X approaches negative infinity of 3 divided by X minus 4. We're going to apply the same property. Limit as x approaches negative infinity of we divided by X minus. Limits X approaches negative infinity of 4. And for this part, the answer doesn't change. We're still dividing 3 by an infinitely large number. The only difference here is that it is a negative number, right? But the limit is still going to be equal to 0. It doesn't matter if we're dividing by an infinitely large. Positive or negative number, any number that is non-zero divided by an infinitely large number, whether positive or negative, leads to the limit of 0. So we get 0 minus 4 again, right? Because the second limit is independent of X, and this gives us -4 again, so we can conclude that the limit as x approaches positive or negative infinity of H of X is going to be equal to -4. Both limits are equal to -4. That will be our final answer and 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.)

f(x) = 2/x − 3

Key Concepts

Limits

Asymptotic Behavior

Graphical Interpretation

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