Math, asked by BrainlyTurtle, 4 months ago

Find Value of the

\displaystyle \lim_{x \to 0} (\frac{3x {}^{2}  + 2}{ {7x}^{2}  + 2}) {}^{ \dfrac{1}{ {x}^{2} } }
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Answers

Answered by BrainlyIAS
163

Question :

Find the value of the  \displaystyle \sf \lim_{x\to 0} \left\{ \dfrac{3x^2+2}{7x^2+2}\right\}^{\dfrac{1}{x^2}}

Solution :

\displaystyle \sf \lim_{x\to 0} \left\{ \dfrac{3x^2+2}{7x^2+2}\right\}^{\dfrac{1}{x^2}}

When we sub. limit directly in the expression , it leads to indeterminant form  \bf 1^{\infty}

Remember the formula ,

\displaystyle \dagger\ \; \sf \orange{\lim_{x \to a} \left[ f(x)\right] ^{[g(x)]} = e^{\displaystyle  \sf \lim_{x\to a} g(x) \left[ f(x)-1 \right]}}

\displaystyle \longrightarrow \sf e^{  \displaystyle \sf \lim_{x \to 0}  \dfrac{1}{x^2} \left\{ \dfrac{3x^2+2}{7x^2+2} - 1 \right\}}

\displaystyle \longrightarrow \sf e^{ \displaystyle \sf \lim_{x \to 0}  \dfrac{1}{x^2}  \left\{ \dfrac{3x^2+2-7x^2-2}{7x^2+2}  \right\}}

\displaystyle \longrightarrow \sf e^{ \displaystyle \sf \lim_{x \to 0}  \dfrac{1}{x^2}  \left\{ \dfrac{-4x^2}{7x^2+2}  \right\}}

\displaystyle \longrightarrow \sf e^{ \displaystyle \sf \lim_{x \to 0}  \left\{ \dfrac{-4}{7x^2+2}  \right\}}

Sub. the limit ,

\displaystyle \longrightarrow \sf e^{ \left\{ \dfrac{-4}{7(0)^2+2}  \right\}}

\displaystyle \longrightarrow \sf e^{ \left\{ \dfrac{-4}{2}  \right\}}

\displaystyle \longrightarrow \sf e^{ \left\{ -2 \right\}}

\displaystyle \longrightarrow \sf \pink{\dfrac{1}{e^2}}

★ ════════════════════ ★

\bullet\ \; \blue{\displaystyle \sf \lim_{x\to 0} \left\{ \dfrac{3x^2+2}{7x^2+2}\right\}^{\dfrac{1}{x^2}} = \dfrac{1}{e^2}}

Answered by Anonymous
118

Answer:

 : \implies \sf e^{-2}

Step-by-step explanation:

To Find:

Value of the \displaystyle \lim_{x \to 0} (\frac{3x {}^{2} + 2}{ {7x}^{2} + 2}) {}^{ \dfrac{1}{ {x}^{2} } }

 \\

Using the formula,

 \bf \dag \displaystyle \lim_{x \to a} [f(x)]^{[g(x)]}=e^{\displaystyle \lim_{x \to a}g(x)[f(x)-1]}

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 : \implies \sf e^{\lim_{x \to a}} \: \dfrac{1}{x^2} \: \bigg[\dfrac{3x^2 + 2}{7x^2 + 2} - 1 \bigg]

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 : \implies \sf e^{\lim_{x \to a}} \: \dfrac{1}{x^2} \bigg[\dfrac{3x^2 + 2 - 7x^2 - 2}{7x^2 + 2} \bigg]

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 : \implies \sf e^{\lim_{x \to a}} \: \dfrac{1}{x^2} \bigg[ \dfrac{-4x^2}{7x^2 + 2} \bigg]

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 : \implies \sf e^{\lim_{x \to a}} \bigg[ \dfrac{-4}{7x^2 + 2} \bigg]

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 : \implies \sf _e \bigg[ \dfrac{4}{7(0)^2 + 2} \bigg]

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 : \implies \sf _e \bigg[ - \dfrac{4}{2} \bigg]

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 : \implies \sf e^{(-2)}

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\implies \large{\underline{\boxed{\bf \dfrac{1}{e^2}}}}

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