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Calculate the limiting value
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$2$
Using a multiplication expression
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{\dfrac {2x ^ {2} - 2x - 3} {x ^ {2} - 1}} \right)$
$ $ Find the limit value of the numerator and denominator separately $ $
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{2x ^ {2} - 2x - 3} \right)\\\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{x ^ {2} - 1} \right)$
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{2x ^ {2} - 2x - 3} \right)\\\lim\limits_{x\to + ∞} \left( x ^ {2} - 1 \right)$
$ $ Calculate the limiting value $ $
$\color{#FF6800}{+ ∞}\\\lim\limits_{x\to + ∞} \left( x ^ {2} - 1 \right)$
$+ ∞\\\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{x ^ {2} - 1} \right)$
$ $ Calculate the limiting value $ $
$+ ∞\\\color{#FF6800}{+ ∞}$
$\color{#FF6800}{+ ∞}\\\color{#FF6800}{+ ∞}$
$ $ Since the limit converging is indedterninate form, convert the formula $ $
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{\dfrac {2x ^ {2} - 2x - 3} {x ^ {2} - 1}} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {2x ^ {2} - 2x - 3} {x ^ {2} - 1} \right)$
$ $ Factorize $ x ^ {2} $ in the formula $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {x ^ {2} \times (2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}})} {x ^ {2} - 1} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {x ^ {2} \times (2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}})} {x ^ {2} - 1} \right)$
$ $ Factorize $ x ^ {2} $ in the formula $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {x ^ {2} \times (2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}})} {x ^ {2} \times (1 - \dfrac {1} {x ^ {2}})} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {\color{#FF6800}{x ^ {2}} \times (2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}})} {\color{#FF6800}{x ^ {2}} \times (1 - \dfrac {1} {x ^ {2}})} \right)$
$ $ Reduce the fraction with $ x ^ {2} $ $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}}} {1 - \dfrac {1} {x ^ {2}}} \right)$
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{\dfrac {2 - \dfrac {2} {x} - \dfrac {3} {x ^ {2}}} {1 - \dfrac {1} {x ^ {2}}}} \right)$
$ $ Calculate the limiting value $ $
$\color{#FF6800}{\dfrac {2 - 2 \times 0 - 3 \times 0} {1 - 0}}$
$\color{#FF6800}{\dfrac {2 - 2 \times 0 - 3 \times 0} {1 - 0}}$
$ $ Solve the formula $ $
$\color{#FF6800}{2}$
$2$
using L'Hospital's theorem
$\lim\limits_{x\to + ∞} \left( \dfrac {2x ^ {2} - 2x - 3} {x ^ {2} - 1} \right)$
$ $ Since the limit value of the numerator and denominator can be an indeterminate form, utilize L'Hospital's theorem $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {\dfrac{d}{dx}(2x ^ {2} - 2x - 3)} {\dfrac{d}{dx}(x ^ {2} - 1)} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {\dfrac{d}{dx}(2x ^ {2} - 2x - 3)} {\dfrac{d}{dx}(x ^ {2} - 1)} \right)$
$ $ Calculate the differentiation $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {4x - 2} {\dfrac{d}{dx}(x ^ {2} - 1)} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {4x - 2} {\dfrac{d}{dx}(x ^ {2} - 1)} \right)$
$ $ Calculate the differentiation $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {4x - 2} {2x} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {4x - 2} {2x} \right)$
$ $ Factorize $ 2 $ in the formula $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {2(2x - 1)} {2x} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {\color{#FF6800}{2}(2x - 1)} {\color{#FF6800}{2}x} \right)$
$ $ Reduce the fraction with $ 2 $ $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {2x - 1} {x} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {2x - 1} {x} \right)$
$ $ Since the limit value of the numerator and denominator can be an indeterminate form, utilize L'Hospital's theorem $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {\dfrac{d}{dx}(2x - 1)} {\dfrac{d}{dx}(x)} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {\dfrac{d}{dx}(2x - 1)} {\dfrac{d}{dx}(x)} \right)$
$ $ Calculate the differentiation $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {2} {\dfrac{d}{dx}(x)} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {2} {\dfrac{d}{dx}(x)} \right)$
$ $ Calculate the differentiation $ $
$\lim\limits_{x\to + ∞} \left( \dfrac {2} {1} \right)$
$\lim\limits_{x\to + ∞} \left( \dfrac {2} {1} \right)$
$ $ Dividing a particular formula by $ 1 $ , the formula remains unchanged $ $
$\lim\limits_{x\to + ∞} \left( 2 \right)$
$\color{#FF6800}{ \lim\limits_{x\to + ∞} } \left( \color{#FF6800}{2} \right)$
$ $ The value of limit is the same as the constant $ $
$\color{#FF6800}{2}$
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