Video Transcript
Find the limit as ๐ฅ tends to zero of ๐ฅ minus four squared minus 16 over ๐ฅ.
The first thing to try here is direct substitution: can we just plug zero in to the expression? We replace ๐ฅ by zero and get zero minus four squared minus sixteen all over zero. Zero minus four is negative four and negative four squared is 16. So simplifying further, we find that itโs just directly substituting zero into the expression, we get the indeterminate form zero over zero.
Weโre going to have to be more clever here. We need to simplify this expression first before substituting in. The first thing we do is to distribute. ๐ฅ minus four squared becomes ๐ฅ squared minus eight ๐ฅ plus 16. We can then cancel the plus 16 and the minus 16. So weโre left with just ๐ฅ squared minus eight ๐ฅ in the numerator.
Both terms in the numerator have a factor of ๐ฅ. And so we can factor the numerator into ๐ฅ times ๐ฅ minus eight. And the factor of ๐ฅ in the numerator cancels with the ๐ฅ in the denominator, leaving us with just ๐ฅ minus eight. As these two expressions are equal, their limits must be equal. And while we saw thatโs direct substitution on the left-hand side gave the indeterminate form zero over zero, direct substitution on the right-hand side โ plugging in ๐ฅ equals zero โ gives negative eight. And hence, the value of the limit that weโre looking for is negative eight.
A reasonable question to ask now is if these two expressions are supposed to be equal, then why did plugging in the zero to the left-hand side give a different answer to plugging in zero to the right-hand side? On the left-hand side, we have the indeterminate form zero over zero and on the right-hand side, we had negative eight. The answer is that in our last step of algebra, where we cancel the factor of ๐ฅ in the numerator with the ๐ฅ in the denominator, we turn something which is undefined when ๐ฅ is zero into something which is defined.
The two expressions are equal for all nonzero values of ๐ฅ. But for ๐ฅ is zero, the left-hand side expression is undefined. As weโre taking the limit as ๐ฅ tends to zero, we donโt care about what the value of the expression is when ๐ฅ is zero. We only care about values of ๐ฅ nearby. The limits then are in fact really equal. And as weโve seen are in fact equal to negative eight.