Final answer:
To compute the limit (x → 2) (√x - √2) / (√(x+1) - √3), we can use the conjugate method. By multiplying the numerator and denominator by (√(x+1) + √3), we can simplify the expression. This results in (√x - √2)(√(x+1) + √3) in the numerator and (√(x+1) - √3)(√(x+1) + √3) in the denominator.
Step-by-step explanation:
To compute the limit lim (x → 2) (√x - √2) / (√(x+1) - √3), we can use the conjugate method. By multiplying the numerator and denominator by (√(x+1) + √3), we can simplify the expression. This results in (√x - √2)(√(x+1) + √3) in the numerator and (√(x+1) - √3)(√(x+1) + √3) in the denominator.
Expanding these expressions gives (√(x^2 + x + √3x + √x) - √2√(x+1) - √2√3) / (x+1 - 3). Simplifying further, (√(x^2 + x + √3x + √x) - 2√(x+1) - √6) / (x-2).
Finally, we substitute x = 2 into this expression to find the limit. Plugging in x = 2 gives (√(4 + 2 + √3(2) + 2) - 2√(2+1) - √6) / (2-2). After further simplification, we get (√(8 + √6) - 2√3 - √6) / 0. This means that the limit is undefined.