Video: Applying Properties of Limits

Video Transcript

Assume that the limit as 𝑥 tends to six of 𝑓 of 𝑥 is equal to three and the limit as 𝑥 tends to six of 𝑔 of 𝑥 is equal to eight. Find the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥.

We need to find the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥. We can break this limit down using the properties of limits. We have the property for the limits of roots of functions. It tells us that the limit as 𝑥 tends to some constant 𝑎 of the 𝑛th root of some function 𝑓 of 𝑥 is equal to the 𝑛th root of the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥. The limit we’re trying to find is the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥. So we have the limit of a square root of a function. We can therefore apply our rule for limits of roots of functions. It tells us that our limit is equal to the square root of the limit as 𝑥 tends to six of 𝑔 of 𝑥 minus 𝑓 of 𝑥.

Now, we can see that we have the limit of a difference of functions since our limit is of 𝑔 of 𝑥 minus 𝑓 of 𝑥. We can apply the rule for the limit of differences of functions, which tells us that the limit as 𝑥 tends to some constant 𝑎 of a difference of functions — so that’s 𝑓 of 𝑥 minus 𝑔 of 𝑥 — is equal to the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 minus the limit as 𝑥 tends to 𝑎 of 𝑔 of 𝑥. We can apply this rule to our limit within the square root, giving us that our limit is equal to the square root of the limit as 𝑥 tends to six of 𝑔 of 𝑥 minus the limit as 𝑥 tends to six of 𝑓 of 𝑥.

Now, we can spot that the limits within our square root have been given to us in the question. We have that the limit as 𝑥 tends to six of 𝑓 of 𝑥 is equal to three and the limit as 𝑥 tends to six of 𝑔 of 𝑥 is equal to eight, giving us that our limit is equal to the square root of eight minus three. Simplifying this, we obtain our solution which is that the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥 is equal to the square root of five.