L'Hopital Rule Intuition

If you get an indeterminate form when $f,g$ cont at $a$ By Linearization, $f(x)\sim f(a)+f^{\prime }(a)(x-a)$ $g(x)\sim g(a)+g^{\prime }(a)(x-a)$ hence, ${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f(a)+f^{\prime }(a)(x-a)}{g(a)+g^{\prime }(a)(x-a)}$ ${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{0+f^{\prime }(a)(x-a)}{0+g^{\prime }(a)(x-a)}$ ${\lim }_{x\to a}\frac{f(x)}{g(x)}={\lim }_{x\to a}\frac{f^{\prime }(a)}{g^{\prime }(a)}$ Since f cont at a, ${\lim }_{x\to a}f(x)=f(a)$ ${\lim }_{x\to a}\frac{f^{\prime }(a)}{g^{\prime }(a)}={\lim }_{x\to a}\frac{f^{\prime }(x)}{g^{\prime }(x)}$