Best solutions to - "Evaluate the following limits. lim(x,y)→(1,2)y−x+1y−x−1"

CoormaBak9

Answered question

2021-01-13

Evaluate the following limits.

lim_{(x, y) \to (1, 2)} \frac{\sqrt{y} - \sqrt{x + 1}}{y - x - 1}

Clara Reese

Skilled2021-01-14Added 120 answers

We have to find the limit:

lim_{(x, y) \to (1, 2)} \frac{\sqrt{y} - \sqrt{x + 1}}{y - x - 1}

It is

\frac{0}{0}

form, since there is square root in numerator so applying rationalization rule multiplying and dividing by

\sqrt{y} + \sqrt{x + 1}

lim_{(x, y) \to (1, 2)} \frac{\sqrt{y} - \sqrt{x + 1}}{(y - x - 1)} = lim_{(x, y) \to (1, 2)} \frac{(\sqrt{y} - \sqrt{x + 1})}{(y - x - 1)} \times \frac{(\sqrt{y} + \sqrt{x + 1})}{(\sqrt{y} + \sqrt{x + 1})}

Since we know the identity,

(a - b) (a + b) = a^{2} - b^{2}

Simplifying further,

lim_{(x, y) \to (1, 2)} \frac{(\sqrt{y})^{2} - (\sqrt{x + 1})^{2}}{(y - x - 1) (\sqrt{y} + \sqrt{x + 1})} = lim_{(x, y) \to (1, 2)} \frac{y - (x + 1)}{(y - x - 1) (\sqrt{y} + \sqrt{x + 1})}

= lim_{(x, y) \to (1, 2)} \frac{(y - x - 1)}{(y - x - 1) (\sqrt{y} - \sqrt{x + 1})}

= lim_{(x, y) \to (1, 2)} \frac{1}{(\sqrt{y} + \sqrt{x + 1})}

= \frac{1}{(\sqrt{2} + \sqrt{1 + 1})}

= \frac{1}{(\sqrt{2} + \sqrt{2})}

= \frac{1}{2 \sqrt{2}}

Hence, value of limits is

\frac{1}{2 \sqrt{2}}

Jeffrey Jordon

Expert2022-04-01Added 2605 answers

Answer is given below (on video)

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