2: Limits and Derivatives
- Page ID
- 208271
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 2 introduces the limit, the fundamental building block of calculus used to describe function behavior near a point. We transition from intuitive graphical estimates to the rigorous \(\epsilon-\delta\) definition and limit laws for algebraic calculation. This framework defines continuity, where a function’s limit must equal its value, and utilizes the Intermediate Value Theorem to prove the existence of roots. Finally, the chapter connects these concepts to the derivative, defining it as the instantaneous rate of change and the slope of a tangent line.
- 2.1: The Tangent and Velocity Problems
- Calculus arose from two geometric puzzles: the tangent problem and the area problem. By using secant lines to approximate a curve’s slope and rectangles to estimate area, we establish the foundation for the limit. This process allows us to transition from approximations to exact values, such as identifying instantaneous velocity as the limit of average velocity over decreasing time intervals.
- 2.2: The Limit of a Function
- The limit of a function \(f(x)\) at a point \(a\) describes the value \(L\) that the function approaches as \(x\) gets arbitrarily close to \(a\), regardless of whether the function is actually defined at \(a\). By examining tables of values or graphs, we can estimate these limits or identify when they fail to exist, such as when a function oscillates wildly or jumps between different values.
- 2.3: Calculating Limits Using the Limit Laws
- While graphs and tables provide an intuitive sense of limits, we primarily use Limit Laws to calculate them analytically. For polynomials and rational functions, this often simplifies to direct substitution, provided the denominator is not zero. When substitution results in an "indeterminate form" like \(0/0\), we use algebraic manipulation—such as factoring, using conjugates, or simplifying complex fractions—to find the limit.
- 2.4: The Precise Definition of a Limit
- The intuitive "getting closer and closer" description of a limit is sufficient for many calculations, but advanced mathematics requires a rigorous, logical foundation. The \(\epsilon\)-\(\delta\) (epsilon-delta) definition provides this by quantifying exactly what "close" means using distances on the \(x\) and \(y\) axes.
- 2.5: Continuity
- A function is continuous at \(a\) if its limit matches its value: \(\displaystyle \lim_{x \to a}f(x)=f(a)\). Discontinuities are classified as removable, jump, or infinite. The Intermediate Value Theorem ensures that a continuous function on \([a, b]\) hits every \(y\)-value between \(f(a)\) and \(f(b)\), providing a key tool for finding roots.
- 2.6: Limits at Infinity; Horizontal Asymptotes
- This section examines function behavior as \(x\) increases or decreases without bound (\(x \to \pm\infty\)). A horizontal asymptote \(y=L\) exists if \( \displaystyle \lim_{x \to \pm\infty} f(x) = L\), signifying the value the function approaches in the far left or right of the graph. For rational functions, these limits are determined by the ratio of the leading terms of the numerator and denominator, while infinite limits at infinity describe functions that grow indefinitely.
- 2.7: Derivatives and Rates of Change
- The derivative \(f'(a)\) measures the instantaneous rate of change and represents the slope of the tangent line at a point. It is the limit of the difference quotient as the interval \(h\) shrinks to zero: \( \displaystyle f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\) Physically, it defines instantaneous velocity, calculating exact speed at a single moment rather than an average.
- 2.8: The Derivative as a Function
- The derivative function \(f'(x)\) assigns the slope of the tangent line at any point \(x\) in the domain. A function is differentiable if this limit exists; points of non-differentiability include corners, cusps, vertical tangents, and any discontinuities. Visually, the graph of \(f'(x)\) tracks the slopes of \(f(x)\), where \(f'(x)=0\) at horizontal tangents and is positive/negative when \(f(x)\) increases or decreases.