The idea of a limit
Consider f(x) = (x² − 4)/(x − 2). At x = 2 it is undefined (0/0). But for x near 2:
x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|
f(x) | 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 |
As x gets closer to 2 from either side, f(x) gets closer to 4. We write
lim (x → 2) f(x) = 4
A limit describes what a function approaches, not its value at the point. Indeed f(x) = x + 2 for x ≠ 2, which approaches 4.
Left and right limits
Left-hand limit (LHL): the value f(x) approaches as x → a from values less than a, written x → a⁻.
Right-hand limit (RHL): as x → a from values greater than a, written x → a⁺.
The limit exists only if LHL = RHL, and then the limit is that common value.
Example. f(x) = |x|/x. For x > 0, f(x) = 1; for x < 0, f(x) = −1. So at 0, RHL = 1 and LHL = −1. They differ, so lim (x → 0) f(x) does not exist.
Algebra of limits
If lim f(x) and lim g(x) exist as x → a:
lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
lim [f(x) g(x)] = lim f(x) × lim g(x)
lim [f(x)/g(x)] = lim f(x) ⁄ lim g(x), provided lim g(x) ≠ 0
lim [k f(x)] = k lim f(x)
Polynomials are well behaved: lim (x → a) p(x) = p(a). Just substitute.
Rational functions: substitute if the denominator isn't zero. If you get 0/0, factorise and cancel the common factor first.
Example. lim (x → 3) (x² − 9)/(x − 3) = lim (x → 3) (x + 3) = 6.
Standard limits
lim (x → a) (xⁿ − aⁿ)/(x − a) = n aⁿ⁻¹
lim (x → 0) sin x / x = 1 (x in radians)
lim (x → 0) (1 − cos x)/x = 0
Example. lim (x → 1) (x¹⁰ − 1)/(x − 1) = 10 × 1⁹ = 10.
Example. lim (x → 0) sin 5x / x = lim 5 × (sin 5x)/(5x) = 5 × 1 = 5.
Example. lim (x → 0) sin 4x / sin 2x = lim [(sin 4x/4x) × 4x] ⁄ [(sin 2x/2x) × 2x] = 4/2 = 2.
Example. lim (x → 0) tan x / x = lim (sin x/x) × (1/cos x) = 1 × 1 = 1.
The second limit is proved geometrically using the areas of a triangle, a sector and a larger triangle in a unit circle, together with the sandwich theorem.
The derivative
The derivative of f at x = a measures the instantaneous rate of change of f, which is the slope of the tangent to the graph at that point:
f′(a) = lim (h → 0) [f(a + h) − f(a)] ⁄ h, if the limit exists.
Calculating this directly is called differentiating from first principles.
Example. f(x) = x².
f′(x) = lim (h → 0) [(x + h)² − x²]/h = lim (2xh + h²)/h = lim (2x + h) = 2x.
Example. f(x) = sin x.
f′(x) = lim [sin(x + h) − sin x]/h = lim [2 cos(x + h/2) sin(h/2)]/h = lim cos(x + h/2) × [sin(h/2)/(h/2)] = cos x.
Physical meaning: if s(t) is the position of an object, its derivative s′(t) is the instantaneous velocity.
Standard derivatives
f(x) | f′(x) |
|---|---|
c (constant) | 0 |
xⁿ | n xⁿ⁻¹ |
sin x | cos x |
cos x | −sin x |
tan x | sec² x |
Rules of differentiation
For differentiable functions u and v:
Sum/difference: (u ± v)′ = u′ ± v′
Constant multiple: (cu)′ = c u′
Product rule: (uv)′ = u′v + uv′
Quotient rule: (u/v)′ = (u′v − uv′) ⁄ v², v ≠ 0
Example. d/dx (3x⁴ − 5x² + 7) = 12x³ − 10x.
Example. d/dx (x² sin x) = 2x sin x + x² cos x.
Example. d/dx [(x + 1)/(x − 1)] = [(1)(x − 1) − (x + 1)(1)] ⁄ (x − 1)² = −2/(x − 1)².
Example. d/dx (1/x) = d/dx (x⁻¹) = −1/x².
Summary
A limit is the value a function approaches; it exists only if LHL = RHL.
For 0/0 forms, factorise or use the standard limits: (xⁿ − aⁿ)/(x − a) → naⁿ⁻¹ and sin x/x → 1.
f′(x) = lim (h → 0) [f(x + h) − f(x)]/h, the slope of the tangent.
d/dx xⁿ = nxⁿ⁻¹; use the product and quotient rules for combinations.