Lesson 5 of 12 · 7 min
Limits of polynomials and rational functions
NCERT §12.3.2, Examples 1 and 2
The pebble splashes into the river at t = 3 s. The average velocity over the last stretch, from t to 3 s, is (44.1 − 4.9t²)/(3 − t). What does it approach as t → 3?
The lesson in notes
In short
For a polynomial f, lim_{x→a} f(x) = f(a): put x = a. Worked: lim_{x→1} (x³ − x² + 1) = 1, lim_{x→3} x(x + 1) = 12 and lim_{x→−1} (1 + x + … + x¹⁰) = 1.
For a rational function g/h with h(a) ≠ 0, the limit is g(a)/h(a). Worked: lim_{x→1} (x² + 1)/(x + 100) = 2/101.
If h(a) = 0 but g(a) ≠ 0, the limit does not exist.
If g(a) = h(a) = 0 (the 0/0 form), write g = (x − a)ᵏ g₁ and h = (x − a)ˡ h₁, cancel the common power, and evaluate again. If k > l the limit is 0; if k < l it is not defined.
Worked: lim_{x→2} (x³ − 4x² + 4x)/(x² − 4) = lim x(x − 2)/(x + 2) = 0, while the upside-down fraction (x² − 4)/(x³ − 4x² + 4x) has no limit at 2.
Worked: lim_{x→2} (x³ − 2x²)/(x² − 5x + 6) = lim x²/(x − 3) = −4. Combining two fractions over a common denominator first, lim_{x→1} [(x − 2)/(x² − x) − 1/(x³ − 3x² + 2x)] = 2.
Cancelling x − a is allowed because the limit only looks at x ≠ a.