Chapter 1
Limits and Continuity

Limits form the basis of calculus: Every single notion of calculus is a limit in one way or another. In this chapter, we will learn what a limit is and how we can calculate it. However, one of the major techniques for finding limits will be postponed until Section 3.2tex4ht

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1.1 Infinite Limits

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Figure 1.1.1:

Consider the function \(f\) defined by the equation \[ f(x)=\frac {1}{(x-1)^{2}} \] The graph of this function is illustrated in Figure 1.1.1.

Note that \(f\) is not defined at \(x=1\) (division by zero is not defined), but let’s consider the values of \(f\) when \(x\) is close to \(1\). Letting \(x\) approach \(1\) from both sides, the corresponding values of \(f\) are given in the following table.

\(x\) \(f(x)\)
0.5 4

1.2 Asymptotes

We say a line is an asymptote of a curve if the distance between the line and the curve approaches zero as the curve (specifically the \(x\) or \(y\) coordinates of the points on the curve) goes to \(+\infty \) or \(-\infty \).

We study three types of asymptotes: (1) vertical, (2) horizontal, and (3) oblique (or inclined or slant).

Vertical Asymptotes

The line \(x=a\) is a vertical asymptote of the graph of \(f\) if \(f(x)\to +\infty \) or \(f(x)\to -\infty \) as \(x\) approaches \(a\) from the left or right.

See Figure 1.2.1.

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Figure 1.2.1:The vertical line \(x=a\) is a vertical asymptote of a curve if the \(y\) coordinates of the points on the curve approach \(+\infty \) or \(-\infty \) as \(x\) approaches \(a\) (from the left or right or both directions).

Chapter 2
Differentiation

The concept of the derivative is one of the two fundamental ideas of calculus, and differentiation is the process of finding the derivative of a function. The derivative of a function is the slope of its tangent line to its graph, and as we will see in this chapter, it is a mathematical tool to measure the rate of change of one quantity relative to another.

The graph of a function \(f\) is shown in Figure 2.0.1. Give a rough sketch of the graph of \(f'.\)

Example 1.

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Figure 2.0.1:

Solution 2. The tangent line to the curve is horizontal at \(x=0\) and \(x=2.\) \[ \begin {cases} x<0 & \text {tangent line makes an acute angle with positive \ensuremath {x} axis}\Rightarrow f'(x)>0\\ 0<x<2 & \text {tangent line makes an obtuse angle with positive \ensuremath {x} axis}\Rightarrow f'(x)<0\\ 2<x & \text {tangent line makes an acute angle with positive \ensuremath {x} axis}\Rightarrow f'(x)>0 \end {cases} \] Specifically, the slope of the tangent line at \(x=-1\) is \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {3}{1}\Rightarrow f'(-1)=3. \] The slope of the tangent line at \(x=1\) is \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {-1}{1}\Rightarrow f'(1)=-1. \] At \(x=3\) \[ m_{\text {tan}}=\frac {\text {rise}}{\text {run}}=\frac {-3}{-1}\Rightarrow f'(3)=3. \]

Chapter 3
Applications of Differentiation

3.1 Concavity and Points of Inflection

Concavity

Recall that if the derivative of a function is positive in an interval, then the function is increasing on that interval, and if the derivative is negative, the function is decreasing (see Chapter 2).

Since \(f^{\prime \prime }(x)\) is the derivative of \(f'(x)\), which is equal to the slope of the tangent line, if \(f^{\prime \prime }(x)>0\), the slope is increasing. This means that moving from left to right, the tangent line turns counterclockwise (Figure 3.1.1a) and the curve bends upward. In this case, we say the function (or its graph) is concave up (the concave side of a curve is its hollow side).

Similarly if \(f^{\prime \prime }(x)\) is negative, the slope is decreasing, the tangent turns clockwise (Figure 3.1.1b), and the curve bends downward. In this case, we say the function (or its graph) is concave down.

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(a) Increasing slope (increasing \(f'\))

(b) Decreasing slope (decreasing \(f'\))

Figure 3.1.1:

Definition 3. A function \(f\) is concave up if \(f'\) is increasing and is concave down if \(f'\) is decreasing.

Theorem 4. If \(f^{\prime \prime }(x)>0\) on an interval \(I\), then \(f\) is concave up on that interval and if \(f^{\prime \prime }(x)<0\), it is concave down.

3.2 L’HÃŽpital’s Rule for Indeterminate Limits

In this section, we learn a powerful method to attack problems on indeterminate forms, called l’HÃŽpital’s rule (also written l’Hospital’s rule).

L’HÃŽpital’s Rule for the Indeterminate Forms \(0/0\) and \(\infty /\infty \)

Assume \(f\) and \(g\) are two functions with \(f(a)=g(a)=0\). Then for \(x\neq a\), we have \[ \frac {f(x)}{g(x)}=\frac {f(x)-\overbrace {f(a)}^{0}}{g(x)-\underbrace {g(a)}_{0}}=\frac {\dfrac {f(x)-f(a)}{x-a}}{\dfrac {g(x)-g(a)}{x-a}} \] Suppose the derivatives \(f'(a)\) and \(g'(a)\) exist and \(g'(a)\neq 0\). Because \[ \lim _{x\to a}\frac {f(x)-f(a)}{x-a}=\lim _{h\to 0}\frac {f(a+h)-f(a)}{h}=f'(a),\qquad (h=x-a) \] and \[ \lim _{x\to a}\frac {g(x)-g(a)}{x-a}=g'(a), \] we get \[ \boxed {\lim _{x\to a}\frac {f(x)}{g(x)}=\frac {f'(a)}{g'(a)}.\tag {i}} \]

The following theorem generalizes the cases where we can use Equation (i). It states that Equation (i) is true under less stringent conditions and is also valid for one-sided limits as well as limits at \(+\infty \) and \(-\infty \).

Exercise 5. Find \(\displaystyle \lim _{x\to 0}\frac {\sin x-\tan x}{x^{3}}.\)