limx→0lim𝑥→0x+29√−29√x

The given expression is:

lim (x→0) [x + √29 - √(29-x)]

To solve this limit, we can use the conjugate method. The conjugate of the expression inside the limit is [x + √29 + √(29-x)].

Step 1: Multiply and divide the expression by its conjugate:

lim (x→0) [(x + √29 - √(29-x)) * (x + √29 + √(29-x))] / (x + √29 + √(29-x))

Step 2: Simplify the numerator using the difference of squares formula (a^2 - b^2 = (a-b)(a+b)):

lim (x→0) [(x + √29)^2 - (√(29-x))^2] / (x + √29 + √(29-x))

Step 3: Simplify the numerator further:

lim (x→0) [x + 58 + x - 29] / (x + √29 + √(29-x))

Step 4: Simplify the expression:

lim (x→0) [2x + 29] / (x + √29 + √(29-x))

Step 5: As x approaches 0, the expression becomes:

[2*0 + 29] / (0 + √29 + √29)

Step 6: Simplify to get the final answer:

29 / (2√29)

So, the limit of the given expression as x approaches 0 is 29 / (2√29).