How do you prove that the limit #x^2 cos(1/x) = 0# as x approaches 0 using the formal definition of a limit?

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Answer 1 · 2016-09-30T17:52:49

Please see below.

Given #epsilon > 0#, choose #delta = sqrt epsilon#.

Now, if #x# is chosen so that #0 < abs(x - 0) < delta#,

then

#abs(x^2cos(1/x) - 0) = abs(x^2) abs( cos(1/x))#

# = x^2 abs( cos(1/x))#

# <= x^2 (1)#

# < (sqrt epsilon)^2 = epsilon#.

That is: if #0 < abs(x - 0) < delta#, then #abs(x^2cos(1/x) - 0) < epsilon#.

So, by definition of limit, #lim_(xrarr0)(x^2cos(1/x) ) = 0#.