Evaluate the following limit:

lim x→1 [(1+x)/(2+x)]^{[(1-√x)/(1-x)]}

(1-√x)/(1-x) = (1-√x)(1+√x)/(1-x)(1+√x)=(1-x)/(1-x)(1+√x)=1/(1+√x)

put this rationalised value in expression

lim x→1 [(1+x)/(2+x)]^[1/(1+√x)]

since this expression does take any indeterminant form put x=1

=(2/3)^1/2

hope this works...

Hi Menka,

The following limit can be written as (2/3)^{lim x→1 (1-√x)/(1-x)} , that is because (1+x)/(2+x) is continuous at x=1.

Now the limit is lim x→1 (1-√x)/[(1-√x)(1+√x)],

Which is  lim x→1 [1/ (1+√x)]

which is 1/2.

Hence the limiting value is (2/3)^1/2.

The solutions to your other questions on functions is ready.

Have to scan it and post it here, as there were some graphs to be made on that.

Will post the scanned copies soon.

All the best.

Regards,

Ashwin (IIT Madras).

lim x->1 (1+x).(1-x^1/2)/(2+x).(1-x)

rationalising with 1+x^1/2

(1+x)/(2+x).(1+x^1/2)

lim x->1 2/6 =1/3