MSM 112: Mathematical Methods II – Limits, Continuity, and Differentiation

Fundamental Concepts and Informal Definitions of Limits

Within the study of calculus, the limit of a function serves as the foundational stone of differential calculus, representing the central idea that distinguishes the discipline from algebra and trigonometry. According to Definition 1, for a function $f(x)$ defined on an open interval surrounding a point $x_0$ (though not necessarily at $x_0$ itself), if $f(x)$ becomes arbitrarily close to a value $L$ for all $x$ sufficiently close to $x_0$, we state that $f$ approaches the limit $L$ as $x$ approaches $x_0$. This is expressed mathematically as $\lim_{x \to x_0} f(x) = L$.

Example 2 illustrates the behavior of the function $f(x) = \frac{x^2 - 1}{x - 1}$ near $x = 1$. While the function is undefined at $x = 1$ due to division by zero, the formula can be simplified for all $x \neq 1$ by factoring the numerator: $f(x) = \frac{(x + 1)(x - 1)}{x - 1} = x + 1$. Consequently, the graph of $f$ is a line $y = x + 1$ with a hole at the point $(1, 2)$. Numerical data supports this limit; as $x$ approaches $1$ from below ($0.9$, $0.99$, $0.999$) and from above ($1.1$, $1.01$, $1.001$), the values of $f(x)$ approach $1.9$, $1.99$, $1.999$ and $2.1$, $2.01$, $2.001$ respectively. Choosing $x$ closer to $1$, such as $0.999999$ or $1.000001$, yields $f(x)$ values of $1.999999$ and $2.000001$. Thus, $\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$. This demonstrates that the limit value does not depend on how the function is defined precisely at $x_0$.

Limit Evaluation Techniques and Identity Functions

Evaluating a limit $\lim_{x \to x_0} f(x)$ can often be achieved through direct substitution by calculating $f(x_0)$. This is particularly applicable when $f(x)$ is an algebraic combination of polynomials and trigonometric functions for which $f(x_0)$ is defined. For example, $\lim_{x \to 2} (5x - 3) = 10 - 3 = 7$, and $\lim_{x \to -2} \frac{3x + 4}{x + 5} = \frac{-6 + 4}{-2 + 5} = -\frac{2}{3}$. In more complex cases, such as $\lim_{x \to 3} \frac{2x^2 + 2x - 5}{x^3 - x^2 + x + 1}$, the result is $\frac{2(3)^2 + 2(3) - 5}{(3)^3 - (3)^2 + 3 + 1} = \frac{18 + 6 - 5}{27 - 9 + 3 + 1} = \frac{19}{22}$.

Specific functions such as Identity and Constant functions have reliable limits at every point. Definition 5 establishes that for an identity function $f(x) = x$, then for any $x_0$, $\lim_{x \to x_0} x = x_0$. Similarly, for a constant function $f(x) = k$, $\lim_{x \to x_0} k = k$ regardless of the value of $x_0$. Examples include $\lim_{x \to 3} x = 3$, and for constant values, $\lim_{x \to -7} 4 = 4$, $\lim_{x \to 2} 4 = 4$, and $\lim_{x \to -13} \frac{7}{3} = \frac{7}{3}$.

The Limit Laws and Algebraic Manipulation

Theorem 6 details the formal Limit Laws provided that $L$, $M$, $c$, and $k$ are real numbers and $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$. The Sum Rule states $\lim_{x \to c} (f(x) + g(x)) = L + M$. The Difference Rule states $\lim_{x \to c} (f(x) - g(x)) = L - M$. The Product Rule states $\lim_{x \to c} (f(x) \cdot g(x)) = L \cdot M$. The Constant Multiple Rule states $\lim_{x \to c} (k \cdot f(x)) = k \cdot L$. The Quotient Rule states $\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{L}{M}$ provided $M \neq 0$. Finally, the Power Rule states that if $r$ and $s$ are integers with no common factor and $s \neq 0$, then $\lim_{x \to x_0} (f(x))^{\frac{r}{s}} = L^{\frac{r}{s}}$, assuming $L^{\frac{r}{s}}$ is a real number (and if $s$ is even, then $L > 0$).

When standard substitution results in a zero denominator, algebraic manipulation such as factoring or rationalization must be employed. Example 8 demonstrates this for $\lim_{x \to 1} \frac{x^2 + x - 2}{x^2 - x}$. Substitution makes the denominator zero, but since the numerator is also zero at $x = 1$, there is a common factor of $(x - 1)$. Simplifying gives $\frac{(x + 2)(x - 1)}{x(x - 1)} = \frac{x + 2}{x}$. Substituting $x = 1$ into the result yields $\frac{1 + 2}{1} = 3$. Example 9 illustrates rationalization for $\lim_{x \to 0} \frac{\sqrt{x^2 + 100} - 10}{x^2}$. By multiplying the numerator and denominator by the conjugate $\sqrt{x^2 + 100} + 10$, we obtain $\lim_{x \to 0} \frac{x^2 + 100 - 100}{x^2(\sqrt{x^2 + 100} + 10)} = \lim_{x \to 0} \frac{1}{\sqrt{x^2 + 100} + 10} = \frac{1}{\sqrt{0 + 100} + 10} = \frac{1}{20}$.

Precise Definition and Limits at Infinity

While informal insight is useful, calculus relies on a precise definition of a limit (Definition 10). We say $\lim_{x \to x_0} f(x) = L$ if for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such that for all $x$, $0 < |x - x_0| < \delta \implies |f(x) - L| < \epsilon$. This rigorous definition is generally reserved for higher-level courses.

Limits at infinity describe the behavior of a function when values in the domain or range outgrow all finite values. The symbol for infinity ($\infty$) is not a real number. For the function $f(x) = \frac{1}{x}$, as $x$ becomes increasingly large and positive, $f(x)$ becomes decreasingly small, approaching $0$. Thus, $\lim_{x \to \infty} \frac{1}{x} = 0$ and $\lim_{x \to -\infty} \frac{1}{x} = 0$. Definition 11 specifies that $\lim_{x \to \infty} f(x) = L$ if for every $\epsilon > 0$ there exists a number $M$ such that $x > M \implies |f(x) - L| < \epsilon$. Similarly, $\lim_{x \to -\infty} f(x) = L$ if for every $\epsilon > 0$ there exists a number $N$ such that $x < N \implies |f(x) - L| < \epsilon$. Theorem 12 confirms that the algebraic Limit Laws for finite limits also apply to limits as $x \to \pm \infty$.

To determine the limit of a rational function as $x \to \pm \infty$, one should divide the numerator and denominator by the highest power of $x$ found in the denominator. In Example 14, where the degrees are equal ($\lim_{x \to \infty} \frac{5x^2 + 8x - 3}{3x^2 + 2}$), dividing by $x^2$ results in $\frac{5 + 0 - 0}{3 + 0} = \frac{5}{3}$. In Example 15, where the degree of the denominator is higher ($\lim_{x \to -\infty} \frac{11x + 2}{2x^3 - 1}$), dividing by $x^3$ produces $\frac{0 + 0}{2 - 0} = 0$.

Infinite Limits and Asymptotes

Infinite limits describe the behavior of functions whose values become arbitrarily large. For $f(x) = \frac{1}{x}$, as $x \to 0^+$, the values grow without bound, written as $\lim_{x \to 0^+} \frac{1}{x} = \infty$. This does not mean the limit exists in a traditional sense, but that the function becomes arbitrarily large and positive. Conversely, as $x \to 0^-$, $\lim_{x \to 0^-} \frac{1}{x} = -\infty$. Example 16 analyzes $y = \frac{1}{x - 1}$, which is a horizontal shift of $y = \frac{1}{x}$ one unit to the right; here $\lim_{x \to 1^+} \frac{1}{x - 1} = \infty$ and $\lim_{x \to 1^-} \frac{1}{x - 1} = -\infty$.

Rational functions exhibit various behaviors near the zeros of their denominators. Sometimes zeros cancel, as in Example 19(i): $\lim_{x \to 2} \frac{(x - 2)^2}{x^2 - 4} = \lim_{x \to 2} \frac{x - 2}{x + 2} = 0$. However, if cancellation is incomplete, infinite limits may occur. For $y = \frac{x - 3}{x^2 - 4}$, as $x \to 2^+$, the limit is $-\infty$, while as $x \to 2^-$, the limit is $\infty$. Since the behavior is inconsistent, $\lim_{x \to 2} \frac{x - 3}{x^2 - 4}$ does not exist. Asymptotes are lines the graph approaches. Example 20 shows that for $y = \frac{x + 3}{x + 2}$, which can be recast as $y = 1 + \frac{1}{x + 2}$, the horizontal asymptote is $y = 1$ and the vertical asymptote is $x = -2$. Example 22 explores oblique asymptotes; for $f(x) = \frac{x^2 - 3}{2x - 4}$, long division yields $\frac{x}{2} + 1 + \frac{1}{2x - 4}$. As $x \to \pm \infty$, the remainder vanishes, leaving the oblique asymptote $y = \frac{x}{2} + 1$.

Limits of Trigonometric, Exponential, and Logarithmic Functions

Definition 23 outlines basic trigonometric limits: $\lim_{x \to c} \sin(x) = \sin(c)$ and $\lim_{x \to c} \cos(x) = \cos(c)$ for all real numbers $c$. Limitations exist for other functions (e.g., $\tan(x)$ is defined for all $c \neq \frac{\pi}{2} + k\pi$). Theorem 24 introduces special limits: $\lim_{x \to 0} \frac{\sin(x)}{x} = 1$ and $\lim_{x \to 0} \frac{1 - \cos(x)}{x} = 0$. These can be extended via Theorem 28: $\lim_{x \to 0} \frac{\sin(nx)}{nx} = 1$. Evaluation often requires creating matching structures, such as in Example 29: $\lim_{x \to 0} \frac{\sin(2x)}{2x^2 + x} = \lim_{x \to 0} \frac{\sin(2x)}{x(2x + 1)} = \lim_{x \to 0} \left( \frac{2 \sin(2x)}{2x} \cdot \frac{1}{2x + 1} \right) = 2(1)(1) = 2$.

Exponential function limits rely on basic facts: $\lim_{x \to \infty} e^x = \infty$ and $\lim_{x \to -\infty} e^x = 0$. Specifically, Theorem 36 notes $\lim_{x \to \infty} (1 + \frac{1}{x})^x = e$ and $\lim_{x \to 0} \frac{e^x - 1}{x} = 1$. For logarithmic functions, finite limits agree with function values if $c > 0$. However, $\lim_{x \to \infty} \log_a(x) = \infty$ and $\lim_{x \to 0^+} \log_a(x) = -\infty$ for $a > 1$. Theorem 41 adds special cases like $\lim_{x \to 1} \frac{\ln(x)}{x - 1} = 1$ and $\lim_{x \to \infty} \frac{\ln(x)}{x} = 0$. Example 79 demonstrates differentiation applications of these, noting $\frac{d}{dx} [\ln(ax + b)] = \frac{a}{ax + b}$.

Continuity and the Continuity Test

A function $f(x)$ is continuous if its graph can be sketched without lifting the pencil. Definition 43 specifies continuity at an interior point $c$ if $\lim_{x \to c} f(x) = f(c)$. For endpoints, one-sided limits must match function values. Definition 44 clarifies that a function is right-continuous at $c$ if $\lim_{x \to c^+} f(x) = f(c)$ and left-continuous if $\lim_{x \to c^-} f(x) = f(c)$. Theorem 46 provides the Continuity Test: a function is continuous at $c$ in its domain if and only if $f(c)$ exists, $\lim_{x \to c} f(x)$ exists (requiring the left-hand limit to equal the right-hand limit), and the limit equals the function value.

Example 47 discusses the unit step function $g(x)$ which is $1$ for $x \geq 0$ and $0$ for $x < 0$. At $x = 0$, $g(0) = 1$, but $\lim_{x \to 0^-} g(x) = 0$ and $\lim_{x \to 0^+} g(x) = 1$. Since the one-sided limits are unequal, $\lim_{x \to 0} g(x)$ does not exist, and the function is discontinuous at $x = 0$. Example 48 features $h(x) = \frac{(x + 2)(x - 1)}{x - 1}$ for $x \neq 1$ and $h(1) = 2$. Here, $\lim_{x \to 1} h(x) = 3$. Although the limit and function value exist, they are not equal, making the function discontinuous at $x = 1$.

Differentiation from First Principles and Basic Rules

Differentiation is the process of finding the differential coefficient of a function. The derivative $f'(x)$ is defined as $\lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$, provided the limit exists. This formula is often called the First Principle. Differentiability requires a "smoothness" condition on the graph; if secant lines through points $P$ and $Q$ do not approach a limiting slope as $Q \to P$, the derivative fails to exist at that point. Common notations include $f'(x)$, $y'$, $\frac{dy}{dx}$, and $Dx f(x)$.

Basic differentiation rules include the Power Rule ($\frac{d}{dx} [x^n] = nx^{n-1}$), Constant Multiple Rule ($\frac{d}{dx} [cf(x)] = cf'(x)$), Constant Rule ($\frac{d}{dx} [c] = 0$), and the Sum and Difference Rules ($\frac{d}{dx} [f(x) \pm g(x)] = f'(x) \pm g'(x)$). The Product Rule is $\frac{d}{dx} [f(x)g(x)] = f(x)g'(x) + g(x)f'(x)$. The Quotient Rule is $\frac{d}{dx} [\frac{f(x)}{g(x)}] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}$. Theorem 60 provides trigonometric derivatives: $\frac{d}{dx} \sin(x) = \cos(x)$ and $\frac{d}{dx} \cos(x) = -\sin(x)$. Related functions follow: $\frac{d}{dx} \tan(x) = \sec^2(x)$, $\frac{d}{dx} \cot(x) = -\csc^2(x)$, $\frac{d}{dx} \sec(x) = \sec(x)\tan(x)$, and $\frac{d}{dx} \csc(x) = -\csc(x)\cot(x)$.

Chain Rule, Implicit Differentiation, and Higher-Order Derivatives

The Chain Rule allows for the differentiation of composite functions $(f \circ g)(x) = f(g(x))$. According to Definition 64, $(f \circ g)'(x) = f'(g(x)) \cdot g'(x)$, or in Leibniz notation, $\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$. Example 66 shows that to differentiate $y = \sin(x^2 + x)$, you set $u = x^2 + x$, find $\frac{dy}{du} = \cos(u)$ and $\frac{du}{dx} = 2x + 1$, resulting in $\frac{dy}{dx} = (2x + 1)\cos(x^2 + x)$.

Implicit functions, where $y$ cannot be easily isolated as $f(x)$, require implicit differentiation. This involves differentiating both sides of an equation with respect to $x$, assuming $y$ is a differentiable function of $x$, and then solving for $\frac{dy}{dx}$. The rule is stated as $\frac{d}{dx} [f(y)] = \frac{d}{dy} [f(y)] \cdot \frac{dy}{dx}$. Example 74 demonstrates this with $2y^2 - 5x^4 - 2 - 7y^3 = 0$, yielding $4y \frac{dy}{dx} - 20x^3 - 21y^2 \frac{dy}{dx} = 0$, then solved as $\frac{dy}{dx} = \frac{20x^3}{4y - 21y^2}$. Logarithmic differentiation is a specific technique where natural logarithms are taken on both sides to simplify products, quotients, or variable exponents (like $y = x^x$) before differentiating implicitly.

Successive Differentiation involves taking derivatives of derivatives. If $y = f(x)$, the second derivative is $f''(x)$ or $\frac{d^2y}{dx^2}$. Higher orders like the third ($\frac{d^3y}{dx^3}$) or $n$-th ($\frac{d^ny}{dx^n}$) can be calculated. Standard formulas for inverse trigonometric derivatives also exist. When $y = \sin^{-1}(x)$, $\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}$. For the general case $y = \sin^{-1} f(x)$, $\frac{dy}{dx} = \frac{f'(x)}{\sqrt{1 - [f(x)]^2}}$. Similar general formulas are defined for $\cos^{-1}$, $\tan^{-1}$, and others, often involving a constant $a$ representing scale, such as $\frac{d}{dx} [\tan^{-1}(\frac{x}{a})] = \frac{a}{a^2 + x^2}$.