evaluate the limit [cant use L'hospital's rule]

Question

1) lim x approaching 2     cos(xpi)[ (cubic root(x+6)) -e^(2-x)]

2) lim x approaching 0     x^2cos((1)/(x^2)) use squeeze theorem

Answer 1

Step 1:

(1)

The expression is  .

.

Step 2:

(2)

The expression is .

The cosine function range between and .

According to squeeze theorem , if for all in an open interval containing c ,

except possibly at c itself, and if ,

then exists and is equal to .

To find and .

Given .

Hence, .

Solution:

(1) .

(2) .