Video Transcript
Find the limit as 𝑥 approaches
zero of sin squared of seven 𝑥 plus three tan squared of three 𝑥 over eight 𝑥
squared.
If we were to try direct
substitution, we would obtain zero over zero, which is undefined. Let’s try to find this limit using
these rules. We’ll also be using the fact that
the limit of a sum of functions is equal to the sum of the limits of the
functions. Hence, we can write our limit as
the sum of these two limits. We notice that we can take a factor
of one-eighth out of the first limit and a factor of three-eighths out of the second
limit.
Next, we notice that both
numerators and both denominators are squares, enabling us to write our limits like
this. Now we can use the fact that the
limit of a square of a function is equal to the square of the limit of the
function. In doing this, we are left with
this. And we notice that our limits look
very similar to the ones we wrote out at the start.
Substituting 𝑎 equals seven into
the first limit rule, we see that our limit on the left must be equal to seven. And substituting 𝑎 equals three
into the second limit rule, we see that our limit on the right must be equal to
three. We obtain one-eighth multiplied by
seven squared plus three-eighths multiplied by three squared. Simplifying this down, we obtain a
solution of 19 over two.
