# Cheat Sheet

MATH 125

## Limits

### Limit Laws

LawEquation
Sum Law$$\lim_{x\to a} \left[f(x)+g(x)\right] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)$$
The Difference Law$$\lim_{x\to a}\left[f(x)-g(x)\right] = \lim_{x\to a}f(x) - \lim_{x\to a}g(x)$$
Constant Multiple Law$$\lim_{x\to a} \left[ cf(x) \right] = c \lim_{x\to a} f(x)$$
Product Law$$\lim_{x\to a}\left[f(x) g(x)\right] = \lim_{x\to a}f(x) \cdot \lim_{x\to a}g(x)$$
Quotient Law$$\lim_{x\to a}\frac{f(x)}{g(x)} = \frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)} \text{ if } \lim_{x\to a}g(x) \neq 0$$
Power Law$$\lim_{x\to a}\left[f(x)\right]^n = \left[\lim_{x\to a}f(x)\right]^n$$
Constant Law$$\lim_{x\to a}c = c$$
Linear Law$$\lim_{x\to a} x = a$$
Exponential Law$$\lim_{x\to a} x^n = a^n$$
Specific Root Law$$\lim_{x\to a} \sqrt[n]{x} = \sqrt[n]{a}$$
General Root Law$$\lim_{x\to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\to a} f(x)}$$

### Limit Case Dynamics

Limit TypeResult
$$c + \infty$$$$\infty$$
$$c - \infty$$$$-\infty$$
$$\infty + \infty$$$$\infty$$
$$-\infty-\infty$$$$-\infty$$
$$\infty-\infty$$Needs further work
$$c\cdot \infty$$, $$c>0$$$$\infty$$
$$c\cdot \infty$$, $$c<0$$$$-\infty$$
$$0 \cdot \infty$$Needs further work
$$\infty \cdot \infty$$$$\infty$$
$$\infty \cdot -\infty$$$$-\infty$$
$$-\infty\cdot-\infty$$$$\infty$$
$$\frac{c}{\pm\infty}$$$$0$$
$$\frac{\pm\infty}{\pm\infty}$$Needs further work
$$\frac{0}{0}$$Needs further work
$$\frac{c}{0}$$Needs further work

### Methods to Solve Limits

• Distribution Substitution Property: if $$f(x)$$ is an algebraic function and $$a$$ is in the domain of $$f$$, then $$\lim_{x\to a} f(x) = f(a)$$. Much of solving for limits is changing the expression of a limit such that the DSP can be used.
• If $$\lim_{x\to a}\frac{P(x)}{Q(x)}$$ is a rational function with $$P(a)=Q(a)=0$$, then $$(x-a)$$ must be a factor of $$P(x)$$ and $$Q(x)$$. Factor it out and remove it from the rational function.
• In $$\frac{0}{0}$$ cases, use rationalization. Multiply numerator/denominator with square root terms by the conjugate.

### Asymptotes

To find horizontal asymptotes of $$f(x)$$, compute $$\lim_{x\to\pm\infty} f(x)$$.

Rules for computing $$x\to\pm\infty$$:

• $$\lim_{x\to\infty} \frac{1}{x^r} = 0$$ for $$r>0$$.
• $$\lim_{x\to\infty} a^x = \infty$$ for $$a > 1$$; $$0$$ for $$a < 1$$.
• $$\lim_{x\to-\infty} a^x = 0$$ for $$a > 1$$; $$\infty$$ for $$a < 1$$.
• Polynomials: identify the term with the highest power; only that term plays the important role.

### Limits to Infinity

• In a polynomial, the highest power domnates the long term behavior.
• In rational functions, the highest power dominates.
• If roots are involved in a rational function, like $$\lim_{x\to -\infty }\frac{\sqrt{x^2+4x}-2x}{x+4}$$, make sure you take into account the negative sign of limits to negative infinity when pulling division by the highest polynomial term into the root.
• $$r^x$$ dominates over every $$x^r$$ (taking into account directionality of the exponential).

Technique: When taking a limit towards infinity or negative infinity for functions involving polynomial terms, find a way to divide all terms by the term by the highest polynomial term. For instance, when calculating $$\lim_{x\to \infty }\frac{5x^4+x^3}{4x^4-9x}$$ one would divide the terms in the numerator and the denominator by $$x^4$$:

$\lim_{x\to \infty }\frac{\frac{5x^4}{x^4}+\frac{x^3}{x^4}}{\frac{4x^4}{x^4}-\frac{9x}{x^4}}=\li _{x\to \:\infty \:}\frac{5+x^{-1}}{4-9x^{-3}}=\frac{5}{4}$

## Derivatives

### Formal Definition

When $$x\to a$$, the slope of the secant line will approach the slope of the tangent line.

$f'(x) = \lim_{x\to a}\frac{f(x) - f(a)}{x-a}$

$$f'(a)$$ is the derivative of $$f$$ at $$x=a$$. The units of the derivative are $$\frac{y \text{ unit}}{x \text{ unit}}$$.

### Derivative Patterns

• When $$f(x)$$ is increasing, $$f'(x)$$ is positive.
• When $$f(x)$$ is decreasing, $$f'(x)$$ is positive.
• When $$f(x)$$ is at a local minima or maxima, $$f'(x) = 0$$.
• When $$f(x)$$ is convex (pointing up, smiling face), $$f'(x)$$ is increasing.
• When $$f(x)$$ is concave (pointing down, frowning face), $$f'(x)$$ is decreasing.

### Fundamental Differentiation Formulas

NameRule
Constant Function Rule$$\frac{d}{dx} c = 0$$
Power Rule$$\frac{d}{dx} x^n = nx^{n-1}$$
Euler Number Rule$$\frac{d}{dx} e^x = e^x$$
Constant Multiplier Rule$$(cf)' = cf'$$
Function Sum Rule$$(f + g)' = f' + g'$$
Function Difference Rule$$(f-g)' = f' - g'$$
Function Product Rule$$(fg)' = fg' + gf'$$
Function Quotient Rule$$\left(\frac{f}{g}\right)' = \frac{gf' - fg'}{g^2}$$
Chain Rule$$\frac{d}{dx} f(g(x)) = f'(g(x))\cdot g'(x)$$

Note that the limit $$\lim_{h\to 0} \frac{\sin(h)}{h} = 1$$.

### Function Specific Differentiation Formulas

FunctionDerivative
$$\ln x$$$$\frac{1}{x}$$
$$a^x$$$$a^x \ln(a)$$
$$\log_a (x)$$$$\frac{1}{x\ln (a)}$$
$$\sin x$$$$\cos x$$
$$\cos x$$$$-\sin x$$
$$\tan x$$$$\frac{1}{\cos^2 x} = \sec^2 x$$
$$\csc x$$$$-\frac{\cos x}{\sin^2 x} = -\cot x \cdot \csc x$$
$$\sec x$$$$\frac{\sin x}{\cos^2 x} = \tan x \cdot\sec x$$
$$\cot x$$$$-\frac{1}{\sin^2 x} = -\csc^2 x$$
$$\sin^{-1} x$$$$\frac{1}{\sqrt{1-x^2}}$$
$$\cos^{-1} x$$$$-\frac{1}{\sqrt{1-x^2}}$$
$$\tan^{-1} x$$$$\frac{1}{1+x^2}$$

### Differentiation Techniques

• Implicit differentiation. If a curve cannot be written explicitly as $$y = ...$$, differentiate both sides of the equation, treat $$y$$ as a function of $$x$$ and thus use chain rule to differentiate, solve for $$y'$$.
• Parametric differentiation. If you are given $$x=x(t)$$ and $$y=y(t)$$, solve for $$\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$.
• Logarithmic differentiation. For complex derivatives as well as derivatives of functions in the form $$f(x)^{g(x)$$, use logarithmic differentiation by taking the log of both sides and implicitly differentiating, then solving for $$y'$$.

## Derivative Applications

### Real Life Rates

Often, in real-world problems we want to understand the dynamics of related rates - as one quantity changes, another desired quantity correspondingly changes.

1. Use geometric tools to establish a mathematical relationship.
2. Identify known variables and derivatives.
3. Implicitly differentiate the mathematical relationship with respect to time or a different appropriate variable.
4. Plug in knowns and solve for the desired derivative.

### Linearization

We can approximate the value of a function by using the line tangent to a nearby convenient point.

Consider a convenient point $$(c_x, c_y)$$, a desired point $$(d_x, d_y)$$, and a function to be evaluated $$f$$.

$d_y = f'(d_x - c_x) + c_y$

### Critical Points

• Critical numbers exist where the derivative is zero or does not exist.
• All extrema are critical numbers, but not all critical numbers are extrema.
• When solving for when the derivative is equal to 0, factor but do not get rid of solutions.
• When solving for absolute extrema within a defined domain $$[a, b]$$, make sure to consider $$x=a$$ and $$x=b$$ as candidates.

### L’Hopital’s Rule

$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$
• Works only for 0/0 or infty/infty.
• When working with roots, it may be easier not to use L’Hopital’s rule.
• Indeterminate limit types require more work.
• Rewrite $$fg$$ as $$\frac{f}{\frac{1}{g}}$$ or $$\frac{g}{\frac{1}{f}}$$ to use L’Hopital’s.
• For $$[\infty - \infty]$$, factor out like $$f - g = f\left(1 - \frac{g}{f}\right)$$
• For exponential limits, use a logarithmic approach.