Question

I could really use some help with this calculus problem.

Find the limit: limx->infinity Xtan(1/x)

Any help would be appreciated

Answer 1

\[\lim_{x \rightarrow \infty} Xtan \frac{ 1 }{ x }\]

Answer 2

Substitute u = 1/x. Thus 1/u = x. Then limit of u as it approaches infinity is the limit of 1/x, which is zero. Then rewrite \[X \tan (1/x) = X *{\sin(1/x)}/\cos(1/x) = {(\sin u)/u}*\cos u\]. Since the limit of u is zero, we use identity:\[\lim_{u \rightarrow 0}{\sin u}/u = 1\] and \[\lim_{u \rightarrow 0} \cos u = \cos(0) = 1\] to show that the answer is .....1.

Answer 3

Where exactly did you get this "u"? Is it just a random variable ?

Answer 4

U is a substitution for 1/x.

Answer 5

& how did you figure that out ?

Answer 6

The limit of 1/x as x approaches infinity is zero. Thus when I substitute 1/x with u I get that the limit of u must be zero.

Answer 7

& we use the trig identities to change tan to sin/cos ?

Answer 8

Can you explain to me why we need to substitute ?

Answer 9

so the answer would be the limit as x approaches infinity = 1 ?