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.)

h(x) = (−5 + (7/x))/(3 – (1/x²))

Accepted Answer

Welcome back everyone to another video. Calculate the limit of H X equals -26 minus 11 divided by X, divided by 13 minus 2 divided by X where x approaches positive and negative infinity. Let's begin with the first limit. Lim is x approaches infinity of negative, 26 minus 11 divided by X, divided by 13 minus 2 divided by X squad. So what we're going to do is simply apply the properties of limits. We can factor out the negative sign. We get negative limit as X approaches infinity, and then we can essentially apply the limit for each part, right? So we have 26 minus 11 divided by X, divided by 13 minus 2 divided by X squad. Now what we have to understand is that we have two fractional terms which contain X. So let's recall that limit as x approaches positive or negative infinity of a divided by X is going to be equal to 0 as long as a is a non-zero number, right? Whenever we divide a number that is non-zero by an infinitely large positive or negative number, the limit is 0. We can then conclude that -11 divided by X is going to approach 0 as x approaches infinity, and -2 divided by X squared is also going to approach 0. It doesn't really matter that X is squared. It basically means that the function grows faster, and we can write our rule as x raised to the power of alpha, right, where alpha is also not equal to 0. It doesn't matter what the exponent is. We just want to make sure that it is. Greater than 0 as well, right, because we must be dividing by an infinitely large positive or negative number but not a fraction. Therefore, alpha cannot be negative. So, from here, we can essentially conclude that our first limit is negative 26 divided by 13. Which is equal to -2. And now if we evaluate our second limit, limit as X approaches negative infinity. We're going to use the same property, we can factor out the the negative sign, and then we're evaluating. The limit of 26 minus 11 divided by X. Divided by 13 minus 2 divided by X squared. Now x approaches negative infinity and the fractional terms are still going to approach 0, right? Therefore, the limit is going to be the same. We get -26 divided by 13, which is -2. So what can we conclude? Well, we can conclude that the limit as X approaches positive or negative infinity of each of X in each case is going to be equal to -2, 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.)

h(x) = (−5 + (7/x))/(3 – (1/x²))

Key Concepts

Limits at Infinity

Simplification of Rational Functions

Graphical Visualization

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