19.1 Limits, One-Sided Limits, Indeterminate Forms & Continuity

19.1 Limits, One-Sided Limits, Indeterminate Forms & Continuity

Mathematical Foundations of the Limit Concept

At the core of differential calculus lies the concept of a limit, which formalizes the intuitive notion of examining the behavior of a function $f(x)$ as its independent variable $x$ approaches a target value $c$, without requiring that $f(c)$ be defined. Formally, we say that the limit of $f(x)$ as $x$ approaches $c$ equals a real number $L$, denoted:

$$\lim_{x \to c} f(x) = L$$

if for every real $\varepsilon > 0$, there exists a corresponding $\delta > 0$ such that for all $x$ satisfying $0 < |x - c| < \delta$, it follows that $|f(x) - L| < \varepsilon$. Crucially, the condition $0 < |x - c|$ explicitly excludes the point $x = c$ itself; limits evaluate the local trajectory of function outputs arbitrarily close to $c$, fundamentally distinguishing limit evaluation from direct function substitution.

One-Sided Limits and the Existence Criterion

Functions frequently exhibit differing trajectories depending on the direction of approach. The left-hand limit denotes the limiting value as $x$ approaches $c$ strictly through values less than $c$ ($x < c$):

$$\lim_{x \to c^-} f(x) = L_1$$

Similarly, the right-hand limit describes the behavior as $x$ approaches $c$ through values greater than $c$ ($x > c$):

$$\lim_{x \to c^+} f(x) = L_2$$

The Two-Sided Limit Existence Criterion establishes that the general (two-sided) limit exists if and only if both one-sided limits exist and converge to the identical finite value:

$$\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = L \quad \text{and} \quad \lim_{x \to c^+} f(x) = L$$

If $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$, the two-sided limit does not exist (DNE). This failure mode is characteristic of jump discontinuities, piecewise definitions, and signum-type functions such as $f(x) = \frac{|x - c|}{x - c}$.

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Fundamental Limit Laws and Linearity

When limits exist finitely, they satisfy algebraic operation rules that permit term-by-term evaluation:

1. Sum and Difference Laws: $\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)$
2. Constant Multiple Law: $\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$ for any scalar $k \in \mathbb{R}$
3. Product Law: $\lim_{x \to c} [f(x) \cdot g(x)] = \left[\lim_{x \to c} f(x)\right] \cdot \left[\lim_{x \to c} g(x)\right]$
4. Quotient Law: $\lim_{x \to c} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$
5. Power and Root Laws: $\lim_{x \to c} [f(x)]^n = \left[\lim_{x \to c} f(x)\right]^n$ and $\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to c} f(x)}$ (with $\lim_{x \to c} f(x) > 0$ when $n$ is even)

For all polynomials and rational functions with non-zero denominators at $x = c$, limits evaluate via direct substitution: $\lim_{x \to c} P(x) = P(c)$.

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The Indeterminate Form $\frac{0}{0}$ & Algebraic Resolution Techniques

When direct substitution into a quotient $\frac{f(x)}{g(x)}$ yields $\frac{0}{0}$, the expression is in an indeterminate form. An indeterminate form conveys no information about whether the limit exists, diverges, or equals a specific real number; instead, it signals that both numerator and denominator share an infinitesimal factor $(x - c)$ that must be eliminated algebraically.

FTCE Mathematics 6-12 candidates must master three primary algebraic resolution techniques:

1. Factoring and Cancellation: For rational expressions where $P(c) = 0$ and $Q(c) = 0$, the Factor Theorem guarantees that $(x - c)$ is a polynomial factor of both. Factoring both polynomials and canceling the common factor $(x - c)$ produces an equivalent expression whose limit can be evaluated directly.
2. Radical Rationalization (Conjugate Multiplication): When expressions involve square roots producing $\frac{0}{0}$, multiply the numerator and denominator by the radical conjugate $(\sqrt{A} + B)$ or $(\sqrt{A} + \sqrt{B})$. Applying the difference-of-squares identity $(\sqrt{A} - B)(\sqrt{A} + B) = A - B^2$ clears the radical and isolates the factor $(x - c)$ for cancellation.
3. Simplification of Complex Fractions: For nested rational expressions, multiply every term in both numerator and denominator by the least common denominator (LCD) of the internal fractions, clearing subsidiary denominators before canceling $(x - c)$.

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Limits Involving Infinity and Asymptotic Geometry

Calculus formalizes Euclidean and rational asymptotes through limits involving infinity:

• Vertical Asymptotes: The line $x = c$ is a vertical asymptote of $y = f(x)$ if at least one of the one-sided limits approaches positive or negative infinity: $$\lim_{x \to c^+} f(x) = \pm\infty \quad \text{or} \quad \lim_{x \to c^-} f(x) = \pm\infty$$ In rational functions, vertical asymptotes arise at isolated zeros of the denominator that do not cancel with numerator zeros.
• Horizontal Asymptotes: The line $y = L$ is a horizontal asymptote if the function approaches $L$ as $x$ grows arbitrarily large in the positive or negative direction: $$\lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L$$ For rational functions $R(x) = \frac{a_n x^n + \dots}{b_m x^m + \dots}$:
  • If $n < m$, $\lim_{x \to \pm\infty} R(x) = 0$, giving horizontal asymptote $y = 0$.
  • If $n = m$, $\lim_{x \to \pm\infty} R(x) = \frac{a_n}{b_m}$, giving horizontal asymptote $y = \frac{a_n}{b_m}$.
  • If $n > m$, the limit is $\pm\infty$; no horizontal asymptote exists (a slant asymptote exists if $n = m + 1$).

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Special Trigonometric Limits

Two foundational limits involving trigonometric ratios cannot be evaluated by algebraic factoring and are established via geometric squeeze arguments:

$$\lim_{x \to 0} \frac{\sin x}{x} = 1 \quad \text{and} \quad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0$$

Using substitution $u = kx$, generalized variants follow immediately:

$$\lim_{x \to 0} \frac{\sin(kx)}{x} = k, \quad \lim_{x \to 0} \frac{\sin(ax)}{\sin(bx)} = \frac{a}{b}, \quad \lim_{x \to 0} \frac{\tan(kx)}{x} = k$$

These relationships are indispensable for establishing the derivatives of $\sin x$ and $\cos x$ from first principles.

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Continuity at a Point & Classification of Discontinuities

Continuity embodies the mathematical guarantee that small variations in input yield small variations in output without abrupt breaks, gaps, or jumps. Formally, a function $f(x)$ is continuous at a point $x = c$ if and only if three criteria are satisfied:

1. $f(c)$ is defined (that is, $c$ is within the domain of $f$).
2. $\lim_{x \to c} f(x)$ exists as a finite real number.
3. $\lim_{x \to c} f(x) = f(c)$ (the limit matches the actual function value).

If any of these conditions fails, $f$ exhibits a discontinuity at $x = c$. Discontinuities are classified into three distinct categories:

• Removable Discontinuity (Hole): $\lim_{x \to c} f(x)$ exists, but either $f(c)$ is undefined or $\lim_{x \to c} f(x) \neq f(c)$. The discontinuity can be removed by defining or redefining $f(c) = \lim_{x \to c} f(x)$.
• Jump Discontinuity: Both one-sided limits $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ exist as finite numbers, but $\lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x)$. The graph displays an abrupt vertical jump of magnitude $|\lim_{x \to c^+} f(x) - \lim_{x \to c^-} f(x)|$.
• Infinite Discontinuity (Essential): One or both one-sided limits diverge to $\pm\infty$. The graph displays an asymptotic rupture along the vertical line $x = c$.

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Limit Evaluation Techniques Matrix

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The Intermediate Value Theorem (IVT) & Root Existence

The Intermediate Value Theorem (IVT) is a foundational existence theorem governing continuous functions on closed bounded intervals:

Intermediate Value Theorem: If $f$ is a continuous real-valued function on the closed interval $[a, b]$, and $u$ is any number strictly between $f(a)$ and $f(b)$ (such that either $f(a) < u < f(b)$ or $f(b) < u < f(a)$), then there exists at least one number $c \in (a, b)$ such that $f(c) = u$.

An essential corollary frequently evaluated on the FTCE examination is Bolzano's Root Existence Theorem: If $f(x)$ is continuous on $[a, b]$ and $f(a)$ and $f(b)$ possess opposite algebraic signs ($f(a) \cdot f(b) < 0$), then $0$ lies strictly between $f(a)$ and $f(b)$. Consequently, there must exist at least one real value $c \in (a, b)$ such that $f(c) = 0$, proving the existence of a real root.

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Worked Exemplar: Algebraic Limit Resolution and IVT Root Verification

Problem:

1. Evaluate the limit algebraically: $$\lim_{x \to 7} \frac{\sqrt{x + 2} - 3}{x^2 - 6x - 7}$$
2. Prove that the continuous function $p(x) = 2x^3 - 9x^2 + 7x + 5$ possesses at least one real zero on the open interval $(1, 3)$.

Step 1: Diagnose the Limit Expression. Test direct substitution into the rational limit expression at $x = 7$: Numerator: $\sqrt{7 + 2} - 3 = \sqrt{9} - 3 = 0$. Denominator: $7^2 - 6(7) - 7 = 49 - 42 - 7 = 0$. The limit exhibits the indeterminate form $\frac{0}{0}$, requiring radical rationalization and polynomial factoring.

Step 2: Apply Radical Conjugate and Factor Denominator. Multiply the numerator and denominator by the conjugate $\sqrt{x + 2} + 3$, and factor the quadratic denominator $x^2 - 6x - 7 = (x - 7)(x + 1)$:

$$\lim_{x \to 7} \frac{(\sqrt{x + 2} - 3)(\sqrt{x + 2} + 3)}{(x - 7)(x + 1)(\sqrt{x + 2} + 3)}$$

Expand the numerator difference of squares:

$$(\sqrt{x + 2})^2 - 3^2 = (x + 2) - 9 = x - 7$$

Substitute this back into the limit:

$$\lim_{x \to 7} \frac{x - 7}{(x - 7)(x + 1)(\sqrt{x + 2} + 3)}$$

Cancel the shared non-zero factor $(x - 7)$ for all $x \neq 7$:

$$\lim_{x \to 7} \frac{1}{(x + 1)(\sqrt{x + 2} + 3)}$$

Step 3: Evaluate by Direct Substitution. Substitute $x = 7$ into the simplified expression:

$$\frac{1}{(7 + 1)(\sqrt{7 + 2} + 3)} = \frac{1}{8 \cdot (3 + 3)} = \frac{1}{8 \cdot 6} = \frac{1}{48}$$

Thus, the algebraic limit converges uniquely to $\frac{1}{48}$.

Step 4: Verify Prerequisite Conditions for the IVT. To prove root existence for $p(x) = 2x^3 - 9x^2 + 7x + 5$ on $(1, 3)$:

• $p(x)$ is a polynomial function, which is continuous on the entire real line $\mathbb{R}$; therefore, $p(x)$ is strictly continuous on the closed interval $[1, 3]$.
• Evaluate $p(x)$ at the interval boundaries $x = 1$ and $x = 3$: $$p(1) = 2(1)^3 - 9(1)^2 + 7(1) + 5 = 2 - 9 + 7 + 5 = 5 > 0$$ $$p(3) = 2(3)^3 - 9(3)^2 + 7(3) + 5 = 2(27) - 9(9) + 21 + 5 = 54 - 81 + 26 = -1 < 0$$

Step 5: Synthesize the Intermediate Value Theorem Conclusion. Because $p(x)$ is continuous on $[1, 3]$ and $p(3) = -1 < 0 < 5 = p(1)$, the target value $u = 0$ lies strictly between $p(3)$ and $p(1)$. By the Intermediate Value Theorem, there must exist at least one real number $c \in (1, 3)$ such that $p(c) = 0$.