Video Transcript
What is the limit as 𝑥 tends to zero of 𝑥 plus two squared minus four all over 𝑥?
Let’s begin by looking at why we simply can’t just substitute 𝑥 is equal to zero into this expression and evaluate it. If we do, we get zero plus two squared minus four all over zero. And we know that anything divided by zero is undefined. So instead, we’re going to look to manipulate our expression somewhat.
A sensible place to start is to begin by distributing the parentheses, remembering that 𝑥 plus two squared is the same as 𝑥 plus two times 𝑥 plus two. We multiply the first term in each bracket. 𝑥 times 𝑥 is 𝑥 squared. We multiply the outer terms. 𝑥 times two is two 𝑥. We then multiply the inner terms. And we get two 𝑥 again. Finally, we multiply the last terms. Two multiplied by two is four. And then we simplify. Two 𝑥 plus two 𝑥 is four 𝑥 and four minus four is zero. So we’re now going to be evaluating the limit as 𝑥 tends to zero of 𝑥 squared plus four 𝑥 over 𝑥.
Notice now we can divide through by 𝑥. 𝑥 squared divided by 𝑥 is 𝑥 and four 𝑥 divided by 𝑥 is four. So we’re now evaluating the limit as 𝑥 tends to zero of 𝑥 plus four. And we can now substitute 𝑥 equals zero into this expression. When we do, we get zero plus four which is equal to four.
So the limit as 𝑥 tends to zero of 𝑥 plus two squared minus four all over 𝑥 is four.
