SIMPLE PROOF FOR L'HOSPITAL'S RULE

Visual idea: chord to tangent

In Calculus, the gradient of the tangent at a point is found by approximating it using the gradient of the chord between two nearby points. As the second point moves closer to the first, the chord “turns into” the tangent.

Linear approximation near \( x = a \)

Consider \( y = f(x) \) and \( y = g(x) \) such that \( f(a) = 0 \) and \( g(a) = 0 \), where \( f \) and \( g \) are differentiable at \( x = a \). — (i)

Then the limit \( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} \) gives the indeterminate form \( \displaystyle \frac{0}{0} \).

By the definition of the derivative at \( x = a \), \( \displaystyle f'(a) \approx \frac{f(x) - f(a)}{x - a} \).
But by (i) \( f(a) = 0 \), this becomes \( \displaystyle f'(a) \approx \frac{f(x)}{x - a} \),
so \( f(x) \approx f'(a)(x - a) \). — (ii)

Similarly, because \( g \) is differentiable at \( x = a \),
we have \( g(x) \approx g'(a)(x - a) \). — (iii)

Plug into the limit

Substitute (ii) and (iii) into \( \displaystyle \lim_{x\to a} \frac{f(x)}{g(x)} \):

\( \displaystyle \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{f'(a)(x - a)}{g'(a)(x - a)} = \frac{f'(a)}{g'(a)} \)

This holds provided \( g'(a) \neq 0 \).

In simple words: very close to \( x = a \), both functions look like straight lines, and L'Hospital's Rule is just taking the ratio of their slopes.