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.
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:
\[
\frac{\sqrt{x} - 7}{x - 49} \cdot \frac{\sqrt{x} + 7}{\sqrt{x} + 7} = \frac{(\sqrt{x} - 7)(\sqrt{x} + 7)}{(x - 49)(\sqrt{x} + 7)}
\]
The numerator simplifies using the difference of squares formula:
\[
(\sqrt{x} - 7)(\sqrt{x} + 7) = x - 49
\]
Thus, the expression becomes:
\[
\frac{x - 49}{(x - 49)(\sqrt{x} + 7)}
\]
We can cancel the common factor \(x - 49\) from the numerator and the denominator:
\[
\frac{1}{\sqrt{x} + 7}
\]
Now, we evaluate the limit as \(x\) approaches 49:
\[
\lim_{x \to 49} \frac{1}{\sqrt{x} + 7} = \frac{1}{\sqrt{49} + 7} = \frac{1}{7 + 7} = \frac{1}{14}
\]
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