Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it. lim x?8 x tan(1/x)

Question

i need detailed solution with working shown.

Answer 1

Since this limit does not have the indeterminate form 0/0 or ∞/∞, you will need to do some rewriting to find it.

 

Notice that x tan (1/x) tends to ∞ · 0, so you must alter the form of the function to match one of the two fractional indeterminate forms. The best way to do this is to put the factor x in the denominator as 1/x:

 

lim_x→∞ x tan (1/x) = lim_x→∞ (tan 1/x) / (1/x).

 

At this point it is useful to replace 1/x by t, so that as x→∞, t→0⁺. The problem then becomes that of

 

lim_t→0+ (tan t)/t = lim_t→0+ (sec² t)/1 = sec² 0 = 1 (the last step because sec² t is continuous for t near 0).