Lesson 3 of 12 · 11 min
Left and right hand limits
NCERT §12.3
The trail to the viewpoint runs flat along the cliff top, 44.1 m above the river, and ends at the edge. One step beyond, the ground is the river bank far below. As a walker's position approaches the edge, what height does the ground approach?
The lesson in notes
In short
Approaching a from values less than a gives the left hand limit, written lim_{x→a⁻} f(x). Approaching from values greater than a gives the right hand limit, lim_{x→a⁺} f(x).
lim_{x→a} f(x) exists only when both one-sided limits exist and are equal; that common number is the limit.
For f(x) = 1 when x ≤ 0 and f(x) = 2 when x > 0, the left hand limit at 0 is 1 and the right hand limit is 2, so the limit at 0 does not exist.
Worked: f(x) = x − 2 for x < 0, 0 at x = 0, and x + 2 for x > 0. At 0 the left hand limit is −2 and the right hand limit is 2, so there is no limit, although f(0) = 0.
Worked: f(x) = x + 2 for x ≠ 1 and f(1) = 0. Both one-sided limits at 1 are 3, so the limit is 3, which differs from f(1). A limit and a value can disagree.