$\lim\limits_{x\to + ∞} \left( \dfrac {2x ^ {2} - 2x - 3} {x ^ {2} - 1} \right)$
$ $ 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 $ $
$\color{#FF6800}{2}$