Question
Use continuity to evaluate the limit. \[ \lim _{x \rightarrow 1} \frac{48+\sqrt{x}}{\sqrt{48+x}} \]
Ask by Logan Norton. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The limit is 7.
Solution
To evaluate the limit
\[
\lim _{x \rightarrow 1} \frac{48+\sqrt{x}}{\sqrt{48+x}},
\]
we can utilize the **continuity** of the functions involved. Both the numerator and the denominator are continuous at \( x = 1 \), meaning we can directly substitute \( x = 1 \) into the expression.
Let's perform the substitution:
\[
\frac{48 + \sqrt{1}}{\sqrt{48 + 1}} = \frac{48 + 1}{\sqrt{49}} = \frac{49}{7} = 7.
\]
Thus, the limit is:
\[
7
\]
**Answer:** 7
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To evaluate the limit \[ \lim_{x \rightarrow 1} \frac{48+\sqrt{x}}{\sqrt{48+x}}, \] we can directly substitute \(x = 1\) into the expression, assuming continuity at that point. Calculating the numerator first: \[ 48 + \sqrt{1} = 48 + 1 = 49. \] Next, we calculate the denominator: \[ \sqrt{48 + 1} = \sqrt{49} = 7. \] Now we can combine the results into the limit: \[ \frac{49}{7} = 7. \] Thus, the limit is \[ \boxed{7}. \]
