Question Video: Finding the Limit of a Rational Function at a Point | Nagwa Question Video: Finding the Limit of a Rational Function at a Point | Nagwa

Question Video: Finding the Limit of a Rational Function at a Point Mathematics • Second Year of Secondary School

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Find lim_(𝑥 → 2) ((𝑥² − 4)²/(4𝑥 − 8))

03:38

Video Transcript

Find the limit as 𝑥 approaches two of 𝑥 squared minus four all squared all divided by four 𝑥 minus eight.

In this question, we’re asked to evaluate the limit as 𝑥 approaches two of a function. And if we were to distribute the square over the parentheses in our numerator, we would see this is a rational function. It’s the quotient of two polynomials. And we know we can attempt to evaluate this by direct substitution. And it’s always a good idea to use direct substitution when it’s possible. So we’ll substitute 𝑥 is equal to two in our function, giving us two squared minus four all squared all divided by four times two minus eight.

But if we evaluate the expression in our numerator and our denominator, we would get zero divided by zero. This is called an indeterminate form. What this means is we can’t determine our limit by using this method, so we’ll have to use some manipulation to rewrite our limit in a form which we can evaluate the limit. And one way of doing this which will work for rational functions is to use what we know about the factor theorem. Substituting 𝑥 is equal to two into the polynomial in our numerator gave us zero. Therefore, by the factor theorem, 𝑥 minus two is a factor of the polynomial in our numerator. The same is also true in our denominator. So let’s try factoring the numerator and the denominator of the function inside of our limit.

Factoring the denominator is easy. We can take out the shared factor of four. Doing this means we can rewrite our limit as the limit as 𝑥 approaches two of 𝑥 squared minus four all squared all divided by four times 𝑥 minus two. We now want to factor our numerator. And to do this, we need to notice inside of our parentheses, we have a difference between two squares. Recall, we know for a difference between two squares, 𝑎 squared minus 𝑏 squared will be equal to 𝑎 minus 𝑏 multiplied by 𝑎 plus 𝑏. And we can see this is true in our numerator. We’ll set 𝑎 equal to 𝑥 and 𝑏 equal to two. So by factoring our numerator using difference between squares, we now have the limit as 𝑥 approaches two of 𝑥 minus two times 𝑥 plus two all squared all divided by four times 𝑥 minus two.

However, we still can’t evaluate this by using direct substitution. We would see we would get a factor of zero in our numerator and a factor of zero in our denominator. So we’re still going to need to do more manipulation. We’ll now distribute the square over the parentheses in our numerator. Doing this, we get 𝑥 minus two all squared multiplied by 𝑥 plus two all squared. And now we’re almost in a position we can evaluate this by using direct substitution. What we want to do is cancel the shared factor of 𝑥 minus two in our numerator and our denominator. If we did this, we would then get an expression which doesn’t have a factor of 𝑥 minus two in our denominator, so we could evaluate this by using direct substitution.

However, it’s worth reiterating why we’re allowed to do this. When we take the limit as 𝑥 approaches two, we’re interested in what happens as 𝑥 gets closer and closer to two. It doesn’t matter what happens when 𝑥 is equal to two. Therefore, we can assume 𝑥 minus two is not equal to zero. And it won’t change the value of our limit. Therefore, the limit given to us in the question will be the same as the limit as 𝑥 approaches two of 𝑥 minus two times 𝑥 plus two all squared all divided by four. And this is the limit as 𝑥 approaches two of a rational function. Or alternatively, it’s the limit as 𝑥 approaches two of a cubic polynomial, so we can do this by using direct substitution.

So we substitute in 𝑥 is equal to two. This gives us two minus two times two plus two all squared all divided by four. And we can see in our numerator, we have a factor of two minus two, which is equal to zero. However, our denominator is equal to four. Therefore, this limit evaluates to give us zero. And this is our final answer. We were able to show by using algebraic manipulation and direct substitution the limit as 𝑥 approaches two of 𝑥 squared minus four all squared all divided by four 𝑥 minus eight is equal to zero.

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