Calc. Practice Problems #14 of 16?

Question

Hello. I have just uploded onto Photobucket 16 calculus practice problems, all of which will be of similar content to my eventual final exam. Now, an answer to these questions would be nice, but is not necessary. Instead, I am concerned with and interested in THE METHOD as to how one could arrive to the correct answer, as well as an EXPLANATION as to why each step needs to be performed in relation to the ones that follow. Anybody who can break down and explain the contents of these problems to me in a SIMPLISTIC way would just be amazing in my book. So thank you very much should you decide to help enlighten me, as I appreciate this more then the words in this post could even begin to convey!

Question #14: http://www.i138.photobucket.com/albums/q271/Link3324/math14.jpg

By the way, This is NOT for any kind of required assignment. It is a guide to work from and to understand before the actual test. Any help would be greatly appreciated. THANK YOU!

Answer 1

for lim x ->pi/4 tan 4x cos 7x
x = pi/4, tan 4x is 0 and cos 7x is sqrt 2/2. Since both are defined, the limit is just the function values, and the limit of the of the product is the product of the limits, 0.
If the function were not defined at that point, the next thing I'd try is simplifying both factors making use of the sum and difference and double angle formulas of trig. Get the tangent in terms of sines and cosines and use L'hospital's rule if necessary.

Answer 2

when x -> pi/2
tan(7x) ---> +infinity
cos(4x) ---> 1

so when x--->pi/2 , the limit of the function is infinity, which means it has no limit