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Varsity · Mathematics · MAT 1

Calculus

From limits to integrals — the language of continuous change. Drag Riemann sums, explore area under curves, and see the Fundamental Theorem come alive.

Limits

Informal definition. We write
lim⁡x→af(x)=L\lim_{x \to a} f(x) = L
and say "the limit of f(x) as x approaches a equals L", if f(x) can be made arbitrarily close to L by taking x sufficiently close to a — but not equal to a. The value of f at a (or even its existence) is irrelevant.
Theorem 1📌 Learn for exam
One-Sided Limits

The left-hand limit as x approaches a from below:

lim⁡x→a−f(x)\lim_{x \to a^-} f(x)

The right-hand limit as x approaches a from above:

lim⁡x→a+f(x)\lim_{x \to a^+} f(x)

The two-sided limit exists if and only if both one-sided limits exist and are equal:

lim⁡x→af(x)=L  ⟺  lim⁡x→a−f(x)=L   and   lim⁡x→a+f(x)=L\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = L \;\text{ and }\; \lim_{x \to a^+} f(x) = L
Theorem 2📌 Learn for exam
Limit Laws

If lim⁡x→af(x)=L\lim_{x\to a}f(x) = L and lim⁡x→ag(x)=M\lim_{x\to a}g(x) = M both exist, then:

lim⁡x→a[f(x)+g(x)]=L+Mlim⁡x→a[f(x)−g(x)]=L−M\lim_{x\to a}[f(x)+g(x)] = L+M \qquad \lim_{x\to a}[f(x)-g(x)] = L-M
lim⁡x→a[f(x)⋅g(x)]=L⋅M\lim_{x\to a}[f(x)\cdot g(x)] = L\cdot M
lim⁡x→af(x)g(x)=LM(M≠0)\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{L}{M} \quad (M\neq 0)
lim⁡x→a[f(x)]n=Ln(n∈Z+)\lim_{x\to a}[f(x)]^n = L^n \quad (n \in \mathbb{Z}^+)
Theorem 3
Fundamental Trigonometric Limit
lim⁡x→0sin⁡xx=1\lim_{x\to 0}\frac{\sin x}{x} = 1

This is proved rigorously using the Squeeze Theorem. It underlies every derivative of a trig function.

▸ One-sided limits — approach x = 2 from both sidestwo sliders, one from each side
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Worked Example

Factoring to resolve an indeterminate form

Find: lim⁡x→2x2−4x−2\lim_{x \to 2}\dfrac{x^2-4}{x-2}

Worked Example

Limit involving sin using the fundamental limit

Find: lim⁡x→0sin⁡(3x)2x\lim_{x \to 0}\dfrac{\sin(3x)}{2x}

Worked Example

One-sided limits — limit does not exist

Find lim⁡x→1f(x)\lim_{x \to 1}f(x) where f(x) = x+1 for x < 1, and 3x−1 for x ≥ 1.

Continuity. A function f is continuous at a if three conditions hold: (i) f(a) is defined, (ii) lim⁡x→af(x)\lim_{x\to a}f(x) exists, and (iii) lim⁡x→af(x)=f(a)\lim_{x\to a}f(x) = f(a). Continuity means the graph has no holes, jumps, or vertical asymptotes at that point.