Evaluate the Limit?

#lim_(xrarr oo) [x(tan(2/x))] #

1 Answer
Mar 20, 2018

The limit is #2#

Explanation:

We can rewrite as

#L = lim_(x-> oo) tan(2/x)/(1/x)#

We see that #tan(2/x)# converges to #0# as #x-> oo# because the larger the value of #x# the smaller the expression within the tangent becomes and the closer #tan(2/x)# becomes to #0#. Also, the limit #lim_(x-> oo) 1/x = 0# is commonly used. Therefore, we may use l;Hosptial's rule.

#L = lim_(x->oo) (-2/x^2 * sec^2(2/x))/(-1/x^2)#

#L = lim_(x-> oo) 2sec^2(2/x)#

The same principal applies with #2/x# as #1/x#: the limit as #x# approaches infinity always remains #0#.

#L = 2sec^2(0)#

#L = 2(1)#

#L = 2#

A graphical verification confirms.

enter image source here

In the above graph, the red curve is #xtan(2/x)# and the blue line is #y = 2#. As you can see, the curve converges onto the line.

Hopefully this helps!