Show that the following limit is true. lim x4 sin 4 First note that we cannot us

Question

Show that the following limit is true. lim x4 sin 4 First note that we cannot use lim X4 sin lim x4 lim sin because the limit as x approaches 0 of sin (1/x) does not exist Instead we apply the squeeze Theorem and so we need to find a function f smaller than g(x) x4 sin(1/x) and a function h bigger than g such that both f(x) and h(x) approach 0. To do this we use our knowledge of the sine function. Because the sine of any number lies between and we can write S Sin Any inequality remains true when multiplied by a positive number. We know that x4 20 for a x and so, multiplying each side of inequalities by x4, we get s x4 sin 1 s as illustrated by the figure. We know that lim X and lim (-x4) Taking Ax) 3 -x 4, g (x) 3x4 sin (1/x), and h(x) 3x4 in the Squeeze Theorem, we obtain m x4 sin

Explanation / Answer

sine function range is [-1,1]

==> -1<= sin(theta) <= 1

here theta = 1/x

==> -1 <= sin(1/x) <= 1

multiply by x4

==> -x4 <= x4sin(1/x) <= x4

apply lim [x->0]

==>lim [x->0] -x4 <= lim [x->0] x4sin(1/x) <= lim [x->0] x4 from squeeze theorem

==> lim [x->0] x4sin(1/x) = 0

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