Determine the following limits.
lim x→∞ (2x − 3) / (4x +...

Question

Determine the following limits.

lim x→∞ (2x − 3) / (4x + 10)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with limits. This problem says to evaluate the following limit the limit as x approaches infinity of the quantity of 7 X plus 2 in quantity, divided by the quantity of 28 X minus 4. We're given 4 possible choices as our answers. For choice A, we have minus 4, for choice B, we have 4. For choice C, we have 1 divided by 4, and for choice D, we have minus 1 divided by 2. Now we're asked to evaluate the limit. As X approaches infinity. Of the quantity of 7 X plus 2 in quantity, divided by the quantity of 28 X minus 4. Now, if we try to evaluate this limit by direct substitution of setting X equal to infinity, we will get infinity divided by infinity, which is an indeterminate form. So, we'll need to evaluate this expression another way. So one way that we can try to evaluate this is to divide both the numerator and denominator by the largest power of X in the denominator, since we have both polynomials on the numerator and denominator. So the largest power of X in the denominator is X-rays to the 1, which is just X, so will divide both the numerator and denominator by X. So doing so, we're going to have our expression equal to the limit. As X approaches infinity. Of the quantity of 7 X divided by X. Plus 2 divided by X in quantity, divided by the quantity of 28 X. Divided by X minus or divided by X. So we can simplify this expression. This is going to be equal to the limit. As X approaches infinity. Of 7 X divided by X is going to give me 7 plus 2 divided by X, and that quantity is divided by the quantity of 28 X divided by X, which is 28. Minus or divided by X. Now both the numerator and denominator, we have the quantity of a constant divided by X. So recall that if we have the limit as X approaches infinity of the quantity of a divided by X-rays to the N. That is going to be equal to 0 or greater than 0. So we can use this information to actually evaluate our limit. So for the limit. As X approaches infinity. Of the quantity of 7 + 2 divided by X in quantity, divided by the quantity of 28 minus 4 divided by X. Here we're gonna actually evaluate the limit. So this is going to be equal to. The limit as X approaches infinity of 7 is just gonna be 7, so we get back to that same constant. So we'll have 7. Plus, and here we have the 2 divided by X, but that's Follows the form of a divided by X to the N, and so we take that limit as X approaches infinity, we're going to get 0. We'll have 7 plus 0. That's gonna be divided by. 28, since it is a constant, minus, and here again, we have a constant divided by X as X purchases infinity, so that's going to evaluate to 0 as well. We'll have minus 0 on the denominator. So simplifying this expression, this is going to be equal to 7 divided by 28. Which is equal to 1 divided by 4. So that is the answer that I'm looking for. And this actually corresponds to answer choice D. So I hope that this explanation has been useful, and I'll see you in the next video.

Determine the following limits.
lim x→∞ (2x − 3) / (4x + 10)

Key Concepts

Limits at Infinity

Rational Functions

Leading Coefficients

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