15. [1/4 Points] DETAILS PREVIOUS ANSWERS SCALCET7 2.3.048 . (( x ) = *+ 1 if* =1 "((x - 2)2 if x z 1' Find the following limits. (If an an lim . A(x ) = 13. Find the limit, if it exists. (If an answer does not exist, enter ONE.) lim_( - TXT) 1. [-/8 Points] EXAMPLE 11 Show that the following limit is true . lim *6 sin (#) = 0 SOLUTION First note that we cannot use Jim * sin () = lim x6 . lim sin() because the limit as x approaches 0 of sin(1/x) does not exist (see this example). Instead we apply the Squeeze Theorem, and so we need to find a function f smaller than s(x) = x sin(1/x) and a unction h bigger than g such that both f(x) and h(x) approach , example r knowledge of the sine function. Because the sine of any number lies between and . we can write Video Example s sin( #) = Any inequality remains true when multiplied by a positive number. We know that x5 2 0 for all x and so, multiplying each side of inequalities by x5, we get s x sin (#) = as illustrated by the figure . We know that lim x = and lim. (-x8) =. Taking A(x) = -x5, g(x) = x sin(1/x), and h(x) = x6 in the Squeeze Theorem, we obtain lim * sin (#) = 0. N A graphing calculator is recommended. se the Squeeze Theorem to show that lim (x2 cos(131x)) = 0. Illustrate by graphing functions R(x) = -x2, g(x) = x2 cos(131x), and h(x) = x2 on the same screen. Let A(x) = -x2, g(x) = x2 cos(13nx), and h(x) = x2. Then[ cos (131x) S 1 v V = [x) v sx cos(131x) = [h(x) v . since lim, f(x) = lim, h(x) = . by the Squeeze Theorem we have lim (x) =[ if 3x - 4 s f(x) s x2 - 3x + 5 for x 2 0, find lim, F(x). Need Help? Watch It f 4x s g(x) s 2x4 - 2x2 + 4 for all x, evaluate lim g(x). ( (x - 2)2 if x = 1' (a) Find the following limits. (If an answer does not exist, enter DNE.) lim _ Rx) = lim . R(x ) =[ ( a ) Evaluate each of the following, if it exists. (If an answer does not exist, enter ONE.) (1) lim- 9(x ) (1) lim+ 9(x) (in) 9(1) (iv) v ) lim , 9( x ) (vi) lim, g(x)