Definition. A function is a relation between a set of inputs (domain) and a set of possible outputs (range) with the property that each input is related to exactly one output.
Note. To determine if a relation is a function, we can use the vertical line test. If a vertical line intersects the graph of the relation at more than one point, then the relation is not a function.
Example. Consider the function \(f(x) = x^2\). This function takes an input \(x\) and returns the square of the input.
Definition. The domain of a function is the set of all real numbers for which the function is defined.
Example. Consider the function \(f(x) = \frac{1}{x}\). The domain of this function is the set of all real numbers except 0 (i.e. \((-\infty, 0)\cup(0, \infty)\). This is because the function is not defined when \(x = 0\).
Definition. The range of a function is the set of all real numbers that the function can take as its output.
Definition. Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.
Example. Consider a right triangle with opposite \(a\), adjacent \(b\), and hypotenuse \(c\). Then the following relationships hold: \begin{align*} \sin(\theta) &= \dfrac{a}{c}, \\ \cos(\theta) &= \dfrac{b}{c}, \\ \tan(\theta) &= \dfrac{a}{b}, \\ \csc(\theta) &= \dfrac{1}{\sin(\theta)}, \\ \sec(\theta) &= \dfrac{1}{\cos(\theta)}, \\ \cot(\theta) &= \dfrac{1}{\tan(\theta)}. \end{align*}
Definition. The secant line of a function is a straight line that intersects the function at given two points.
Definition. The tangent line of a function is a straight line that intersects the function at given one point.
Definition. The average velocity of an object is the change in position of the object divided by the time interval over which the change occurs. Mathematically, the average velocity is given by \[ \text{Average velocity} = \frac{\text{Change in position}}{\text{Time interval}}. \]
Note. If you graph the position function, where your y-axis is the position of the object, and the x-axis is the time, the average velocity is the slope of the line that connects two points on the graph (i.e. the slope of the secant line).
Definition. The instantaneous velocity of an object is the velocity of the object at a given instant in time. Mathematically, the instantaneous velocity is given by \[ \text{Instantaneous velocity} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}, \] where \(\Delta x\) is the change in position of the object and \(\Delta t\) is the time interval over which the change occurs.
Note. If you graph the position function, where your y-axis is the position of the object, and the x-axis is the time, the instantaneous velocity is the slope of the tangent line at a given point on the graph.
To approximate the instantaneous velocity of an object at a given point, we can use the average velocity over a small-time interval that contains the point. The smaller the time interval, the better the approximation.
Definition. The limit of a function is the value that the function approaches as the input (variable) approaches a given value.
Example. Consider the function \(f(x) = x^2\). Then \(\lim\limits_{x \to 2} f(x) = 4\). This is because as \(x\) approaches 2, the value of \(f(x)\) approaches 4.
Note. If \(f(x)\) is a constant function (i.e. the value of the function will not change as the input changes), then the limit of the function is the constant value.
Example. Consider the function \(f(x) = 3\) defined all real numbers. Then \(\lim\limits_{x \to 2} f(x) = 3\). In general, if \(f(x) = k\) for all \(x\), then \[ \lim\limits_{x \to c} f(x) = k. \]
Definition. The left limit of a function is the value that the function approaches as the input approaches a given value from the left side. We denote the left limit as \[ \lim_{x \to c^-} f(x), \] where the superscript \(^-\) indicates that the input is approaching from the left side.
Definition. The right limit of a function is the value that the function approaches as the input approaches a given value from the right side. \[ \lim_{x \to c^+} f(x), \] where the superscript \(^+\) indicates that the input is approaching from the right side.
Definition. The limit of a function exists if the left and right limits of the function exist and are equal. Mathematically, the limit of a function exists if \[ \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x). \]
Example. Consider the function \(f(x) = x^2\). Then \(\lim\limits_{x \to 2} f(x) = 4\). This is because the left and right limits of the function are both 4.
Definition. The law of limits states that the limit of a sum is the sum of the limits, the limit of a product is the product of the limits, and the limit of a quotient is the quotient of the limits.
Theorem. Given that \(\lim\limits_{x \to c} f(x) = L<\infty\) and \(\lim\limits_{x \to c} g(x) = M<\infty\), then \begin{align*} \lim_{x \to c} \left(k\cdot f(x)\right) &= k\cdot L \\ \lim_{x \to c} \left(f(x) + g(x)\right) &= L + M, \\ \lim_{x \to c} \left(f(x) \cdot g(x)\right) &= L \cdot M, \\ \lim_{x \to c} \left(\frac{f(x)}{g(x)}\right) &= \frac{L}{M}, \quad \text{if } M \neq 0. \end{align*}
Example. Consider the functions \(f(x) = x^2\) and \(g(x) = x\). Then \(\lim\limits_{x \to 0} f(x) = 0\) and \(\lim\limits_{x \to 0} g(x) = 0\). Therefore, \(\lim_{x \to 0} f(x) + g(x) = 0 + 0 = 0\).
Definition. A function is continuous at a point if the limit of the function at that point exists and is equal to the value of the function at that point. Mathematically, a function is continuous at a point \(c\) if \[ \lim_{x \to c} f(x) = f(c). \]
Note. The immediate consequence of being continuous are
Definition. A function is discontinuous at a point if the limit of the function at that point does not exist or is not equal to the value of the function at that point.
Definition. A function is continuous from the left at a point if the limit of the function from the left side exists and is equal to the value of the function at that point. Mathematically, a function is continuous from the left at a point \(c\) if \[ \lim_{x \to c^-} f(x) = f(c). \]
Definition. A function is continuous from the right at a point if the limit of the function from the right side exists and is equal to the value of the function at that point. Mathematically, a function is continuous from the right at a point \(c\) if \[ \lim_{x \to c^+} f(x) = f(c). \]
Removable Discontinuity. A removable discontinuity occurs when the limit of the function at a point exists, but the value of the function at that point is not equal to the limit. Mathematically, a removable discontinuity occurs when \[ \lim_{x \to c} f(x) = L \quad \text{and} \quad f(c) \neq L. \]
Jump Discontinuity. A jump discontinuity occurs when the limit of the function at a point from the left side is not equal to the limit of the function at that point from the right side. Mathematically, a jump discontinuity occurs when \[ \lim_{x \to c^-} f(x)\text{ exists and } \lim_{x \to c^+} f(x)\text{ exists, but } \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x). \]
Infinite Discontinuity. An infinite discontinuity occurs when the limit of the function at a point is infinite. Mathematically, an infinite discontinuity occurs when \[ \lim_{x \to c} f(x) = \pm \infty, \text{ or }\lim_{x \to c^-} f(x) = \pm \infty,\text{ or }\lim_{x \to c^+} f(x) = \pm \infty. \]
Definition. An elementary function is a function of one variable which is the composition of a finite number of operations (\(+, -, \times, \div\)), exponential, logarithms, constants, polynomials, trigonometric functions, inverse trigonometric functions, and roots.
Note. \(\sqrt[n]{x}\), \(\log_a(x)\), \(a^x\), \(\sin(x)\), \(\cos(x)\), \(\tan(x)\) are elementary functions on their domains.
Theorem. If \(f(x)\) and \(g(x)\) are elementary functions, then the following functions are also elementary functions: \begin{align*} f(x) \pm g(x), \\ f(x) \times g(x), \\ f(x) \div g(x) = \dfrac{f(x)}{g(x)}, \\ f\circ g = f(g(x)), \\ \end{align*}
Example. \(f(x) = \log_{5}(x^2 + 3) - e^{\sin(x)}\) and \(g(x) = \dfrac{\sqrt{\sin(e^x)}}{2x^4 - x^2 + 1}\) are elementary functions.
Important. If \(f(x)\) is an elementary function, then \(f(x)\) is continuous on its domain.
Theorem. If \( f(x) \) and \( g(x) \) are continuous at \( x = c \), then
Definition. An indeterminate form is a mathematical expression that cannot be evaluated using the standard rules of arithmetic.
Examples. \(0^0\), \(\infty - \infty\), \(\dfrac{0}{0}\), \(\dfrac{\infty}{\infty}\), \(0 \cdot \infty\), \(\infty^0\).
Important. If you encounter an indeterminate form, you may need to use factoring, times the conjugate, or other methods to evaluate the limit.
Definition. The conjugate of a complex number \(a + b\) is \(a - b\). The conjugate of a complex number \(a - b\) is \(a + b\).
Example. \[ \begin{align} \lim_{x\to 7}\dfrac{\sqrt{x+9}-4}{x-7} &= \lim_{x\to 7}\left(\dfrac{\sqrt{x+9}-4}{x-7}\cdot\dfrac{\sqrt{x+9}+4}{\sqrt{x+9}+4}\right) \\ &= \lim_{x\to 7}\dfrac{x+9-16}{(x-7)(\sqrt{x+9}+4)} \\ &= \lim_{x\to 7}\dfrac{x-7}{(x-7)(\sqrt{x+9}+4)} \\ &= \lim_{x\to 7}\dfrac{1}{\sqrt{x+9}+4} \\ &= \dfrac{1}{\sqrt{7+9}+4} \\ &= \dfrac{1}{8}. \end{align} \]
Example 1. \[ \begin{align} \lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{11}{x^2 + 7x - 18} \right) &= \lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{11}{(x + 9)(x - 2)} \right) \\ &= \lim_{x \to 2} \left( \frac{x + 9}{(x + 9)(x - 2)} - \frac{11}{(x + 9)(x - 2)}\right) \\ &= \lim_{x \to 2} \left( \frac{x + 9 - 11}{(x + 9)(x - 2)} \right) \\ &= \lim_{x \to 2} \left( \frac{x - 2}{(x + 9)(x - 2)} \right) \\ &= \lim_{x \to 2} \left( \frac{1}{x + 9} \right) \\ &= \frac{1}{2 + 9} \\ &= \frac{1}{11}. \end{align} \]
Example 2. \[ \begin{align} \lim_{x \to \frac{\pi}{4}} \dfrac{\cot(4x)}{\csc(4x)} &= \lim_{x \to \frac{\pi}{4}} \left(\dfrac{\dfrac{1}{\tan(4x)}}{\dfrac{1}{\sin(4x)}}\right) \\ &= \lim_{x \to \frac{\pi}{4}} \left(\dfrac{\sin(4x)}{\tan(4x)}\right) \\ &= \lim_{x \to \frac{\pi}{4}} \left(\dfrac{\sin(4x)}{\dfrac{\sin(4x)}{\cos(4x)}}\right) \\ &= \lim_{x \to \frac{\pi}{4}} \cos(4x) \\ &= \cos\left(4\left(\dfrac{\pi}{4}\right)\right) \\ &= \cos(\pi) \\ &= -1. \end{align} \]
Example. \[ \begin{align} \lim_{x \to 5^-} \frac{4x^2 - 27x + 35}{x^2 - 25} &= \lim_{x \to 5^-} \frac{(4x - 7)(x - 5)}{(x + 5)(x - 5)} \\ &= \lim_{x \to 5^-} \frac{4x - 7}{x + 5} \\ &= \frac{4(5) - 7}{5 + 5} \\ &= \frac{13}{10} \\ &= 1.3. \end{align} \]
Sometimes, you are required to use several ways to evaluate a limit. Here is one example using the conjugate and simplify the rationals.
Example. \[ \begin{align} \lim_{\theta \to \frac{\pi}{2}} (3 \sec(\theta) - 3 \tan(\theta)) &= \lim_{\theta \to \frac{\pi}{2}} \left(3\dfrac{1}{\cos(\theta)} - 3\dfrac{\sin(\theta)}{\cos(\theta)}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3 - 3\sin(\theta)}{\cos(\theta)}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3(1 - \sin(\theta))}{\cos(\theta)}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3(1 - \sin(\theta))\cdot (1 + \sin(\theta))}{\cos(\theta)\cdot (1 + \sin(\theta))}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3(1 - \sin^2(\theta))}{\cos(\theta)\cdot (1 + \sin(\theta))}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3\cos^2(\theta)}{\cos(\theta)\cdot (1 + \sin(\theta))}\right) \\ &= \lim_{\theta \to \frac{\pi}{2}} \left(\dfrac{3\cos(\theta)}{1 + \sin(\theta)}\right) \\ &= \dfrac{3\cos\left(\dfrac{\pi}{2}\right)}{1 + \sin\left(\dfrac{\pi}{2}\right)} \\ &= \dfrac{3(0)}{1 + 1} \\ &= 0. \end{align} \]
Note. Keep in mind that \(\sin^2(\theta) + \cos^2(\theta) = 1\).
Example. \[ \begin{align} \lim_{x \to \infty} \dfrac{2x^3 + 2x + 1}{5x^3 +3x^2 + 7} &= \lim_{x \to \infty} \dfrac{\left(2x^3 + 2x + 1\right)\cdot \frac{1}{x^3}}{(5x^3 +3x^2 + 7)\cdot \frac{1}{x^3}} \\ &= \lim_{x \to \infty} \dfrac{\dfrac{2x^3}{x^3} + \dfrac{2x}{x^3} + \dfrac{1}{x^3}}{\dfrac{5x^3}{x^3} + \dfrac{3x^2}{x^3} + \dfrac{7}{x^3}} \\ &= \lim_{x \to \infty} \dfrac{2 + \dfrac{2}{x^2} + \dfrac{1}{x^3}}{5 + \dfrac{3}{x} + \dfrac{7}{x^3}} \\ &= \dfrac{2 + 0 + 0}{5 + 0 + 0} \\ &= \dfrac{2}{5}. \end{align} \]