Questions: Find lim as x approaches 49 of (√x-7)/(x-49)

Transcript text: Find $\lim _{x \rightarrow 49} \frac{\sqrt{x}-7}{x-49}$

Solution

Solution Steps

To find the limit, we can use algebraic manipulation to simplify the expression. Specifically, we can multiply the numerator and the denominator by the conjugate of the numerator, which is $\sqrt{x} + 7$. This will help eliminate the square root and simplify the expression, allowing us to evaluate the limit as $x$ approaches 49.

Step 1: Simplify the Expression

To find the limit $\lim _{x \rightarrow 49} \frac{\sqrt{x}-7}{x-49}$, we start by simplifying the expression. We multiply the numerator and the denominator by the conjugate of the numerator, $\sqrt{x} + 7$, to eliminate the square root:

\[ \frac{\sqrt{x} - 7}{x - 49} \cdot \frac{\sqrt{x} + 7}{\sqrt{x} + 7} = \frac{(\sqrt{x} - 7)(\sqrt{x} + 7)}{(x - 49)(\sqrt{x} + 7)} \]

Step 2: Apply the Difference of Squares

The numerator simplifies using the difference of squares formula:

\[ (\sqrt{x} - 7)(\sqrt{x} + 7) = x - 49 \]

Thus, the expression becomes:

\[ \frac{x - 49}{(x - 49)(\sqrt{x} + 7)} \]

Step 3: Cancel Common Factors

We can cancel the common factor $x - 49$ from the numerator and the denominator:

\[ \frac{1}{\sqrt{x} + 7} \]

Step 4: Evaluate the Limit

Now, we evaluate the limit as $x$ approaches 49:

\[ \lim_{x \to 49} \frac{1}{\sqrt{x} + 7} = \frac{1}{\sqrt{49} + 7} = \frac{1}{7 + 7} = \frac{1}{14} \]