Question #73434

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Answer 1 · 2017-03-16T13:42:57

#=1#

#lim_(x to oo) x sin (1/x)#

let # z = 1/x#

#color(red)( lim_(z to 0) (sin (z))/z = 1)#

The red expression is a well known fundamental result.

Answer 2 · 2017-03-16T13:59:36

# lim_(x to oo) x sin (1/x) = 1#

We want to find:

# lim_(x to oo) x sin (1/x) #

Let #theta = 1/x#, then as #x rarr oo => theta rarr 0#

And the limit can be rewritten, and evaluated;

# lim_(x to oo) x sin (1/x) = lim_(theta to 0) 1/theta sin theta#
# " " = lim_(theta to 0) (sin theta)/theta#
# " " = 1#

Where we used a fundamental trigonometry calculus limit:

# lim_(theta to 0) (sin theta)/theta =1 #