20 Evaluate the following limit Use LHospitals Rule where po

20) Evaluate the following limit. Use L\'Hospital\'s Rule where possible. If limit results in an 20) indeterminate form where L\'Hospital\'s Rule cannot be applied directly, manipulate the of the limit that L\'Hospital\'s Rule can be applied directly. lim (tan (x) cos(x) x (TU/2)

Solution

y = tan(x)^cos(x)

ln(y) = cos(x)*ln(tan(x))

Now we need to find lim cos(x)*ln(tan(x)):

lim cos(x)*ln(tan(x)) =

lim ln(tan(x))/(1/cos(x)) = (-> Use L\'Hopital\'s rule here)

lim ((1+tan²x)/tan(x))/(sin(x)/cos²x) =

lim cos²x(1+tan²x)/(sinx.tanx) =

lim 1/(sin²x/cosx) =

lim cos(x)/sin²x =

cos(/2)/sin²(/2) =

0/1 =0

Therefore:

lim ln(y) = 0   -> lim y = e⁰ = 1

So we have:

lim (tan(x))^cos(x) = 1