Question Video: Finding the Limit of a Combination of Root and Rational Functions at a Point by Direct Substitution

Find lim_(𝑥 ⟶ 8) √((7𝑥 + 7)/(3𝑥 − 3)).

02:16

Video Transcript

Find the limit as 𝑥 approaches eight of the square root of seven 𝑥 plus seven divided by three 𝑥 minus three.

We’re asked to evaluate the limit. And the first thing we should do whenever we’re asked to evaluate a limit is see if we can evaluate this by direct substitution. To decide whether we can evaluate the following limit by direct substitution, we need to check the function in our limit.

First, we see the inner function is a rational function; it’s seven 𝑥 plus seven all divided by three 𝑥 minus three. And we know we can evaluate the limit of a rational function by direct substitution as long as our denominator does not evaluate to give us zero. And in this case, we can see this is true. Since our limit is as 𝑥 is approaching eight, we would substitute 𝑥 is equal to eight into our rational function. And in our denominator, we would get three times eight minus three which is 24 minus three which we can calculate is equal to 21, which is not zero.

But this is not just the limit of a rational function; we’re also taking the square root of this expression. We know we can evaluate the limit of power functions by direct substitution. And by using our laws of exponent, we know taking the square root of a number is the same as raising it to the power of one-half. So, taking the square root of this rational function is the same as raising it to the power of one-half. So, we can evaluate the rational function by direct substitution and the power function by direct substitution.

This means we’re evaluating the limit of the composition of two functions which we can evaluate by direct substitution. This means we can evaluate our limit by direct substitution. So, we’re now ready to evaluate this limit by direct substitution. We just substitute 𝑥 is equal to eight in the function inside of our limit. This gives us the square root of seven times eight plus seven divided by three times eight minus three.

And now, we can just start evaluating this expression. We have seven times eight plus seven is equal to 63 and three times eight minus three is equal to 21. So, we have the square root of 63 divided by 21. And we know 63 divided by 21 simplifies to give us three. So, this limit evaluated to give us the square root of three.

Therefore, in this question, we were asked to evaluate the limit of the square root of a rational function. And we were able to prove that we can evaluate this limit by direct substitution by using our rules. And then, we just evaluated this limit by substituting 𝑥 is equal to eight into the function inside of our limit. We got that it was equal to the square root of three.

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