CalcRef

Topics · 18 formulas

Limits & Continuity

Limit definitions, laws, special limits, and continuity theorems.

Limit Definition (Epsilon-Delta)

lim⁡x→af(x)=L  ⟺  ∀ϵ>0,∃δ>0:0<∣x−a∣<δ⇒∣f(x)−L∣<ϵ\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon

The formal epsilon-delta definition of a limit. For every epsilon greater than zero, there exists a delta such that f(x) is within epsilon of L whenever x is within delta of a.

Conditions: f(x) must be defined on an open interval containing a, except possibly at a itself.

Definitions

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Left-Hand Limit

lim⁡x→a−f(x)=L\lim_{x \to a^-} f(x) = L

The limit of f(x) as x approaches a from the left (from values less than a).

Conditions: f(x) must be defined on an open interval (c, a) for some c < a.

Definitions

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Right-Hand Limit

lim⁡x→a+f(x)=L\lim_{x \to a^+} f(x) = L

The limit of f(x) as x approaches a from the right (from values greater than a). A two-sided limit exists if and only if both one-sided limits exist and are equal.

Conditions: f(x) must be defined on an open interval (a, d) for some d > a.

Definitions

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Squeeze Theorem

g(x)≤f(x)≤h(x) and lim⁡x→ag(x)=lim⁡x→ah(x)=L⇒lim⁡x→af(x)=Lg(x) \leq f(x) \leq h(x) \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \Rightarrow \lim_{x \to a} f(x) = L

If f(x) is squeezed between g(x) and h(x) near a, and g and h have the same limit L, then f also has limit L.

Continuity & Theorems

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Limit of a Sum

lim⁡x→a[f(x)+g(x)]=lim⁡x→af(x)+lim⁡x→ag(x)\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

The limit of a sum equals the sum of the limits, provided both limits exist.

Conditions: Both lim f(x) and lim g(x) must exist.

Limit Laws

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Limit of a Product

lim⁡x→a[f(x)⋅g(x)]=lim⁡x→af(x)⋅lim⁡x→ag(x)\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

The limit of a product equals the product of the limits, provided both limits exist.

Conditions: Both lim f(x) and lim g(x) must exist.

Limit Laws

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Limit of a Quotient

lim⁡x→af(x)g(x)=lim⁡x→af(x)lim⁡x→ag(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero.

Conditions: Both limits must exist and lim g(x) ≠ 0.

Limit Laws

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Limit of a Power

lim⁡x→a[f(x)]n=[lim⁡x→af(x)]n\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit, for any positive integer n.

Conditions: lim f(x) must exist and n is a positive integer.

Limit Laws

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Limit of a Constant Multiple

lim⁡x→a[c⋅f(x)]=c⋅lim⁡x→af(x)\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

A constant factor can be pulled out of a limit.

Conditions: lim f(x) must exist and c is a constant.

Limit Laws

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Limit of sin(x)/x

lim⁡x→0sin⁡xx=1\lim_{x \to 0} \frac{\sin x}{x} = 1

One of the most important special limits in calculus. Often proved using the Squeeze Theorem with geometric arguments.

Special Limits

Open formula

Limit of (cos x - 1)/x

lim⁡x→0cos⁡x−1x=0\lim_{x \to 0} \frac{\cos x - 1}{x} = 0

A special limit related to the derivative of cosine at x = 0.

Special Limits

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Limit Definition of e

lim⁡n→∞(1+1n)n=e\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

The number e (≈ 2.71828) defined as a limit. Equivalently, lim(x→0) (1+x)^(1/x) = e.

Special Limits

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L'Hopital's Rule

If 00 or ±∞±∞:lim⁡x→af(x)g(x)=lim⁡x→af′(x)g′(x)\text{If } \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule: When a limit gives an indeterminate form 0/0 or ∞/∞, the limit equals the ratio of the derivatives (if that limit exists).

Conditions: The limit must produce 0/0 or ±∞/±∞. g'(x) ≠ 0 near a. The derivative limit must exist (or be ±∞).

Special Limits

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Continuity Definition

f is continuous at a  ⟺  lim⁡x→af(x)=f(a)f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a).

Continuity & Theorems

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Intermediate Value Theorem

f continuous on [a,b],  f(a)<N<f(b)⇒∃ c∈(a,b):f(c)=Nf \text{ continuous on } [a,b],\; f(a) < N < f(b) \Rightarrow \exists\, c \in (a,b) : f(c) = N

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists at least one c in (a,b) where f(c) = N. Often used to show a root exists.

Conditions: f must be continuous on the closed interval [a, b].

Continuity & Theorems

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Extreme Value Theorem

f continuous on [a,b]⇒f attains an absolute max and min on [a,b]f \text{ continuous on } [a,b] \Rightarrow f \text{ attains an absolute max and min on } [a,b]

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on that interval.

Conditions: f must be continuous and the interval must be closed and bounded.

Continuity & Theorems

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Limits at Infinity

lim⁡x→∞anxn+⋯bmxm+⋯={0n<man/bmn=m±∞n>m\lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

For rational functions as x→∞, compare the degrees of numerator (n) and denominator (m) to determine the limit.

Continuity & Theorems

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Infinite Limits (Vertical Asymptotes)

lim⁡x→af(x)=±∞⇒x=a is a vertical asymptote\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a \text{ is a vertical asymptote}

If f(x) approaches ±∞ as x approaches a, then the line x = a is a vertical asymptote of f.

Continuity & Theorems

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