Evaluate the following limit algebraically.

Question

lim (sqrt 4+x-2)/x
x approaches 0

Answer 1

you must mean
lim ( √(4+x) - 2)/x

multiply top and bottom by √(4+x) + 2

lim ( √(4+x) - 2)/x
= lim ( √(4+x) - 2)/x * (√(4+x) + 2)/(√(4+x) + 2)
= lim (4+x - 4)/(x(√(4+x) + 2))
= lim x/(x(√(4+x) + 2))
= lim 1/(√(4+x) + 2) , as x ---> 0 (for eachof the above lines as well)
= 1/(√(4+0) + 2)
= 1/4