Hello Jennifer!

unfortunately, you cannot multiply both the denominator and numerator by (3 + sqrt t). 

When evaluating the limits in calculus, you have to expand the values given to simplify your final answer as the limit approaches a certain number given i.e (t --> 9). in this case, t approaches 9, hence when the eqn is simplified, you substitute all values of t into the eqn. [e.g: the value of t --> 9 is substituted into (root t ) to find the value of the final answer. ]

step 1: 

at this step, firstly you start expanding the value (9 - t ) = (3 + sqrt t) (3 - sqrt t). e.g  (3*3 = 9  and sqrt t * sqrt t = t ), therefore you are expanding the given eqn.

illustration:

step 2:

you are expanding to eliminate  (3 - root t) at the denominator to the (3 - root t) at the numerator.

step 3:

substitute (t --> 9 ) value into root t. [e.g (3 + root t = 3 + root(9) ) = 6 ].

that are the rules of calculating limits. For additional information relating to limits, click this link: https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-2/v/introduction-to-imits-hd  or https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new   

- Hope this helps!

~Galaxy.

Wow! Thank you so much!

I really appreciate how you explained all the steps and why they are done that way - that's the way my brain works. Furthermore, I did not realize that t could be expanded to sqrt t and sqrt t, but this makes sense since sqrt t * sqrt t = t. 

Thanks again,

Jen

Alright, no worries! I'm glad that was helpful :)