Question
Use algebra and the properties of limits as needed to find the given limit. If the limit does not exist, say so. \( \lim _{x \rightarrow 2} \frac{7 x+5}{2 x+6} \)
Ask by Osborne Schneider. in the United States
Feb 21,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The limit is 1.9.
Solution
Calculate the limit \( \lim_{x \rightarrow 2} \frac{7x+5}{2x+6} \).
Evaluate the limit by following steps:
- step0: Evaluate the limit:
\(\lim _{x\rightarrow 2}\left(\frac{7x+5}{2x+6}\right)\)
- step1: Rewrite the expression:
\(\frac{\lim _{x\rightarrow 2}\left(7x+5\right)}{\lim _{x\rightarrow 2}\left(2x+6\right)}\)
- step2: Calculate:
\(\frac{19}{\lim _{x\rightarrow 2}\left(2x+6\right)}\)
- step3: Calculate:
\(\frac{19}{10}\)
The limit of \( \frac{7x+5}{2x+6} \) as \( x \) approaches 2 is \( \frac{19}{10} \) or 1.9.
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Simplify this solution Extra Insights
To find the limit \( \lim_{x \rightarrow 2} \frac{7x + 5}{2x + 6} \), we can simply substitute \( x = 2 \) into the function: First, substitute: \[ \frac{7(2) + 5}{2(2) + 6} = \frac{14 + 5}{4 + 6} = \frac{19}{10} \] Since the function is continuous at \( x = 2 \), the limit exists and is equal to \( \frac{19}{10} \). So, we can conclude that: \[ \lim_{x \rightarrow 2} \frac{7x + 5}{2x + 6} = \frac{19}{10} \]
