Question
()) If \( \lim _{x \rightarrow 6} f(x)=10 \) and \( \lim _{x \rightarrow 6} h(x)=-5 \), what is \( \lim _{x \rightarrow 6}[f(x)+h(x)]^{2} \) ? ()) Simplify your levels answer.
Ask by Lane Schmidt. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\lim_{x \to 6} [f(x) + h(x)]^2 = 25
\]
Solution
To find the limit of \([f(x) + h(x)]^2\) as \(x\) approaches 6, follow these steps:
1. **Find the Limits of \(f(x)\) and \(h(x)\):**
\[
\lim_{x \to 6} f(x) = 10
\]
\[
\lim_{x \to 6} h(x) = -5
\]
2. **Add the Limits:**
\[
\lim_{x \to 6} [f(x) + h(x)] = \lim_{x \to 6} f(x) + \lim_{x \to 6} h(x) = 10 + (-5) = 5
\]
3. **Square the Result:**
\[
\lim_{x \to 6} [f(x) + h(x)]^2 = \left(\lim_{x \to 6} [f(x) + h(x)]\right)^2 = 5^2 = 25
\]
**Final Answer:**
\[
\lim_{x \to 6} [f(x) + h(x)]^2 = 25
\]
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Simplify this solution Extra Insights
To find \( \lim _{x \rightarrow 6}[f(x)+h(x)]^{2} \), we start by using the limits you've given. We know that \( \lim _{x \rightarrow 6} f(x) = 10 \) and \( \lim _{x \rightarrow 6} h(x) = -5 \). Using the property of limits, we can combine them: \[ \lim _{x \rightarrow 6}[f(x) + h(x)] = \lim _{x \rightarrow 6} f(x) + \lim _{x \rightarrow 6} h(x) = 10 + (-5) = 5. \] Now we need to take the square of this limit: \[ \lim _{x \rightarrow 6}[f(x)+h(x)]^{2} = [\lim _{x \rightarrow 6}(f(x) + h(x))]^{2} = 5^{2} = 25. \] So, \( \lim _{x \rightarrow 6}[f(x)+h(x)]^{2} = 25 \).
