Link Search Menu Expand Document

Cheat Sheet

MATH 125


Limit Laws

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
\(c + \infty\)\(\infty\)
\(c - \infty\)\(-\infty\)
\(\infty + \infty\)\(\infty\)
\(\infty-\infty\)Needs further work
\(c\cdot \infty\), \(c>0\)\(\infty\)
\(c\cdot \infty\), \(c<0\)\(-\infty\)
\(0 \cdot \infty\)Needs further work
\(\infty \cdot \infty\)\(\infty\)
\(\infty \cdot -\infty\)\(-\infty\)
\(\frac{\pm\infty}{\pm\infty}\)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.


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 \(x^4\):

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


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

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

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

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.


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.