Evaluate the following limit algebraically.

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

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

To evaluate the limit algebraically, we can simplify the expression first.

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

To simplify, we can rationalize the numerator by multiplying the expression by its conjugate. The conjugate of sqrt(4+x) - 2 is sqrt(4+x) + 2.

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

Expanding the numerator, we get:

lim [(sqrt(4+x))^2 - 4] / x(sqrt(4+x) + 2) as x approaches 0.

Simplifying further, we have:

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

The numerator becomes 0 after simplification:

lim 0 / x(sqrt(4+x) + 2) as x approaches 0.

Now, we can see that the limit is 0, since the numerator approaches 0 and the denominator is non-zero. Therefore, the algebraic evaluation of the given limit is 0.