L'Hôpital's Rule & Linear Approximation
L'Hôpital's Rule resolves 0/0 or ∞/∞ limits by differentiating numerator and denominator separately. Linear approximation uses a tangent line to estimate function values near the point of tangency.
L'Hôpital's Rule
If lim f(x)/g(x) gives 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x), provided the second limit exists. You can reapply it repeatedly if you still get an indeterminate form.
L'Hôpital's Rule only applies to indeterminate forms — using it on a limit that isn't 0/0 or ∞/∞ gives a wrong answer.
Worked Example
Find lim(x→0) sin(x)/x using L'Hôpital's Rule.
- Direct substitution gives 0/0 — indeterminate, so L'Hôpital's Rule applies.
- Differentiate top and bottom separately: d/dx[sin x]=cos x, d/dx[x]=1.
- New limit: lim(x→0) cos(x)/1 = cos(0)/1 = 1.
Answer: 1
Linear approximation (tangent line approximation)
Near x=a, f(x) ≈ f(a) + f'(a)(x-a). This is just the equation of the tangent line, used as a nearby estimate for values you can't compute exactly.
Linear approximation tends to overestimate where the curve is concave down (curve dips below the tangent line) and underestimate where the curve is concave up.
Key terms (2)
- L'Hôpital's Rule
- For 0/0 or ∞/∞ forms, lim f/g = lim f'/g'.
- Linear approximation
- f(x) ≈ f(a) + f'(a)(x-a), using the tangent line at x=a to estimate nearby values.
Practice Quiz
Find lim(x→0) (eˣ - 1)/x.