Limit And Continuity

Rationalize the numerator:

%5Cfrac%7B%5Csqrt%7B1%2Bx%7D-1%7D%7Bx%7D%0A%3D%20%5Cfrac%7B(%5Csqrt%7B1%2Bx%7D-1)(%5Csqrt%7B1%2Bx%7D%2B1)%7D%7Bx(%5Csqrt%7B1%2Bx%7D%2B1)%7D.

Simplify using

(%5Csqrt%7B1%2Bx%7D-1)(%5Csqrt%7B1%2Bx%7D%2B1)%20%3D%20(1%2Bx)-1%20%3D%20x.

So the expression becomes:

%5Cfrac%7Bx%7D%7Bx(%5Csqrt%7B1%2Bx%7D%2B1)%7D%20%3D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1%2Bx%7D%2B1%7D.

Now take the limit as

x%20%5Cto%200

%5Clim_%7Bx%5Cto%200%7D%20%5Cfrac%7B1%7D%7B%5Csqrt%7B1%2Bx%7D%2B1%7D%0A%3D%20%5Cfrac%7B1%7D%7B1%2B1%7D%0A%3D%200.5.

Thus,

%5Cboxed%7B0.5%7D

Chapter Questions
1 MCQs