Topic
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Question
Evaluate the following limit using L'Hospital's rule.
\[ \lim _{x \rightarrow 0^{+}}(2 x)^{\left(\frac{1}{6 \ln (8 x)}\right)} \]
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2. Sea \( f(x)=4 x^{2}-x+5 \) y \( g(x)=2 x^{3}-x \). Encuentra la derivada de \( h(x)=\frac{f(x)}{g(x)} \)
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4. (10 points) Find all relative and absolute extrema of the function
\[ f(t)=\frac{1}{2} t^{4}+\frac{10}{3} t^{3} \]
with domain \( (-\infty,+\infty) \). Then the same function with domain \( [-2,+\infty) \)
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Question
Evaluate the following limit using L'Hospital's rule.
\[ \lim _{x \rightarrow 0^{+}}(6 x)^{\left(\frac{1}{7 \ln (3 x)}\right)} \]
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Question
Evaluate the following limit using L'Hospital's rule.
\[ \lim _{x \rightarrow 1}\left[\frac{2}{6 \ln (x)}-\frac{2}{6 x-6}\right] \]
Provide an exact answer.
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1. Dadas las funciones \( f(x)=x^{3}+2 x-1 \) y \( g(x)=x^{2}+1 \), calcula la derivada de \( h(x) \)
\( \frac{f(x)}{g(x)} \).
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Question
Evaluate the following limit using L'Hospital's rule.
\[ \lim _{x \rightarrow 0^{-}}\left(\frac{-7}{x}+\frac{1}{\sin (6 x)}\right) \]
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An object with initial temperature \( 160^{\circ} \mathrm{F} \) is submerged in large tank of water whose temperature is \( 70^{\circ} \mathrm{F} \).
Find a formula for \( F(t) \), the temperature of the object after \( t \) minutes, if the cooling constant is
\( k=-1.4 \)
\( F(t)=\square \)
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3. Sea \( f(x)=x^{5}+3 x^{2} \) y \( g(x)=4 x-x^{2} \). Determina la derivada de \( h(x)=f(x) \cdot g(x) \)
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Question
Evaluate the following limit using L'Hospital's rule.
\[ \lim _{x \rightarrow 12^{+}}\left[\frac{168}{5 x^{2}-85 x+300}-\frac{2 x}{5 x-60}\right] \]
Provide an exact answer.
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