Mathematics RU

Exams · First-year calculus

Limits

The forms 0/0 and ∞/∞: factor, cancel, divide by the highest power.

How to solve it

Limits

The limit $\lim_{x \to a} f(x)$ is the number the values of the function approach as $x$ approaches $a$. Often it is enough to substitute $a$. Trouble starts when substitution gives nonsense like $\frac{0}{0}$ or $\frac{\infty}{\infty}$.

Step by step

  1. Substitute the point. If you get a number, that is the answer.
  2. If you get $\frac{0}{0}$, factor the numerator and the denominator, cancel the common factor and substitute again.
  3. If $x \to \infty$ and you get $\frac{\infty}{\infty}$, divide the numerator and the denominator by the highest power of $x$: everything divided by $x$ tends to zero.
  4. For a ratio of polynomials of the same degree the limit at infinity is the ratio of the leading coefficients.
The leading coefficient of the numerator. The leading coefficient of the denominator. The degrees are equal; if the denominator's degree is higher, the limit is $0$. An example with $\frac{0}{0}$: $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = \lim_{x \to 2} (x + 2) = 4$.

Common mistakes

  • Writing that $\frac{0}{0}$ equals $0$ or $1$. It is not a number but a signal to transform the expression.
  • Dividing only some of the terms by the highest power.
  • Comparing the wrong degrees: if the numerator's degree is lower than the denominator's, the limit at infinity is zero.

Example

Mathematics

Chapters

Introduction

  1. 0 What mathematics is

I Numbers

  1. 1 Counting
  2. 2 Zero and minus
  3. 3 Primes
  4. 4 Divisibility
  5. 5 Fractions
  6. 6 Root two

II Algebra

  1. 7 Letters
  2. 8 Equations
  3. 9 Functions and graphs
  4. 10 Quadratic equations
  5. 11 Systems and inequalities
  6. 12 Powers and logarithms
  7. 13 Sequences
  8. 14 Polynomials
  9. 15 Complex numbers

III Geometry

  1. 16 Euclid's Elements
  2. 17 The triangle
  3. 18 Pythagoras' theorem
  4. 19 The circle and π
  5. 20 Area and volume
  6. 21 Trigonometry
  7. 22 Vectors
  8. 23 Symmetry
  9. 24 Conic sections

IV Calculus

  1. 25 Limits
  2. 26 The derivative
  3. 27 Extrema
  4. 28 The integral
  5. 29 The number e
  6. 30 Series
  7. 31 Differential equations
  8. 32 Several variables
  9. 33 Fourier series
  10. 34 Complex analysis

V Linear algebra

  1. 35 Matrices
  2. 36 Gaussian elimination
  3. 37 Vector spaces
  4. 38 Eigenvectors
  5. 39 Least squares and SVD

VI Structures

  1. 40 Groups
  2. 41 Remainders and ciphers
  3. 42 Rings and fields
  4. 43 Galois theory
  5. 44 Zeta and curves

VII Chance & data

  1. 45 Combinatorics
  2. 46 Graphs
  3. 47 Probability
  4. 48 Random variables
  5. 49 Statistics
  6. 50 Markov and information

VIII Foundations

  1. 51 Logic and sets
  2. 52 Infinities
  3. 53 Rigorous analysis
  4. 54 Measure and integral
  5. 55 Functional analysis
  6. 56 Gödel and Turing

IX Horizons

  1. 57 Topology
  2. 58 Non-Euclidean geometries
  3. 59 Chaos and fractals
  4. 60 The frontier