Problem 25 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the expression as (x, y) approaches (1, 2): $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$ Answer: The limit of the given expression as (x, y) approaches (1, 2) is $$\frac{1}{2\sqrt{2}}$$.

Step 1: Observe the given limit expression

Observe the given limit expression: $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$

Step 2: Perform a change of variables

To simplify the expression, perform a change of variables: let \(u = x + 1\) and \(v = y\). Now, as \((x, y) \rightarrow (1, 2)\), we have \((u, v) \rightarrow (2, 2)\). Rewrite the limit expression using \(u\) and \(v\): $$\lim _{(u, v) \rightarrow(2, 2)} \frac{\sqrt{v}-\sqrt{u}}{v-u}$$

Step 3: Apply the conjugate to the numerator

To further simplify the expression, multiply the numerator and denominator by the conjugate of the numerator, which is \(\sqrt{v} + \sqrt{u}\). This can help to remove square roots from the expression: $$\lim _{(u, v) \rightarrow(2, 2)} \frac{(\sqrt{v}-\sqrt{u})(\sqrt{v}+\sqrt{u})}{(v-u)(\sqrt{v}+\sqrt{u})}$$

Step 4: Simplify the expression

Now, simplify the expression by calculating the product of the numerators and updating the denominators: $$\lim _{(u, v) \rightarrow(2, 2)} \frac{v-u}{(v-u)(\sqrt{v}+\sqrt{u})}$$

Step 5: Simplify the expression further

The expression can be further simplified as the \((v-u)\) term cancels out from the numerator and denominator: $$\lim _{(u, v) \rightarrow(2, 2)} \frac{1}{\sqrt{v}+\sqrt{u}}$$

Step 6: Substitute the limit values

Now that the expression is simplified, we can substitute the values for \(u\) and \(v\) as they approach \((2, 2)\) to evaluate the limit: $$\lim _{(u, v) \rightarrow(2, 2)} \frac{1}{\sqrt{v}+\sqrt{u}} = \frac{1}{\sqrt{2}+\sqrt{2}} = \frac{1}{2\sqrt{2}}$$ So, the limit of the given expression as \((x, y) \rightarrow (1, 2)\) is \(\frac{1}{2\sqrt{2}}\).