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How do you find the limit of #xsin(pi/x)# as x approaches infinity?

Answer 1

Samantha Adkison

Use #lim_(thetararr0)sintheta/theta# and some algebra.

#xsin(pi/x) = sin(pi/x)/(1/x)#

# = pi/pi sin(pi/x)/(1/x)#

# = pi sin(pi/x)/(pi/x)#

With #theta = pi/x#, we have

#lim_(xrarroo) xsin(pi/x) = lim_(xrarroo) pi sin(pi/x)/(pi/x)#

# = pi lim_(xrarroo) sin(pi/x)/(pi/x)#

# = pi lim_(thetararr0) sintheta/theta#

# = pi#

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Answer 2

Cameron Alexander

To find the limit of xsin(pi/x) as x approaches infinity, we can use the squeeze theorem. By applying the squeeze theorem, we can establish that the limit is equal to zero.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.