Problem 26 - 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: Determine the value of the following limit: $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$ Answer: The value of the given limit is $$\frac{\sqrt{2}}{4}.$$

Step 1: Analyzing the given limit

Given the limit: $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1}$$ We want to find the limit of the function as \((x,y) \rightarrow (1,2)\). Let's try direct substitution:

Step 2: Direct substitution

Substitute \((x,y)=(1,2)\) into the function: $$\frac{\sqrt{2}-\sqrt{1+1}}{2-1-1} = \frac{\sqrt{2}-\sqrt{2}}{0}$$ We can see that we have a form \(\frac{0}{0}\), which is indeterminate. Direct substitution didn't work. Now, let's try an algebraic method called Rationalization.

Step 3: Rationalization

First, let's rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator: $$\frac{\sqrt{y}-\sqrt{x+1}}{y-x-1} * \frac{\sqrt{y}+\sqrt{x+1}}{\sqrt{y}+\sqrt{x+1}} = \frac{y-(x+1)}{(y-x-1)(\sqrt{y}+\sqrt{x+1})}$$ Now, simplify the expression:

Step 4: Simplify the expression

Simplify the expression obtained after rationalization: $$\frac{y-x-1}{(y-x-1)(\sqrt{y}+\sqrt{x+1})} = \frac{1}{\sqrt{y}+\sqrt{x+1}}$$ At this point, we have a simplified expression that we should now be able to evaluate the limit for.

Step 5: Direct substitution (again)

Substitute \((x,y)=(1,2)\) into the simplified expression: $$\lim _{(x, y) \rightarrow(1,2)} \frac{1}{\sqrt{y}+\sqrt{x+1}} = \frac{1}{\sqrt{2}+\sqrt{1+1}} = \frac{1}{\sqrt{2}+\sqrt{2}} = \frac{1}{2\sqrt{2}}$$ Now, simplify the denominator to obtain the final answer:

Step 6: Final simplification

To further simplify, we can multiply the numerator and denominator by \(\sqrt{2}\): $$\frac{1}{2\sqrt{2}} * \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{4}$$ So, the limit is equal to the final result: $$\lim _{(x, y) \rightarrow(1,2)} \frac{\sqrt{y}-\sqrt{x+1}}{y-x-1} = \frac{\sqrt{2}}{4}$$