Limit of \( \displaystyle \frac{x - 5}{\sqrt{x - 1} - 2} \) as \( x \to 5 \)

Problem 1.220 · medium

1. \[ \lim_{x \to 5^+}\left(\frac{x - 5}{\sqrt{x - 1} - 2}\right) \]
   limit algebraStart with the limit of the given function. Multiply the numerator and denominator by the conjugate of the denominator.✓ Proved
2. \[ = \lim_{x \to 5^+}\left(\sqrt{x - 1} + 2\right) \]
   simplify simplify cancelSimplify the denominator using the difference of squares. Combine the constants in the denominator. Cancel the common factor (x - 5).✓ Proved
3. \[ = 4 \]
   substitution simplify simplify limitSubstitute x = 5 into the expression. Simplify the expression inside the square root. Calculate the square root. Final result.✓ Proved