limit superior

Examples of limit superior in the following topics:

• Absolute Convergence and Ratio and Root Tests

  • The usual form of the test makes use of the limit, $L = \lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|$.
  • if $L = 1$ or the limit fails to exist, then the test is inconclusive, because there exist both convergent and divergent series that satisfy this case.
  • For a series $\sum_{n=1}^\infty a_n$, the root test uses the number $C = \limsup_{n\rightarrow\infty}\sqrt[n]{ \left|a_n \right|}$, where "lim sup" denotes the limit superior, possibly ∞.
  • if $C = 1$ and the limit approaches strictly from above, then the series diverges;

• Limit of a Function

  • The notion of a limit has many applications in modern calculus.
  • In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.
  • If both of these limits are equal to $L$ then this can be referred to as the limit of $f(x)$ at $p$.
  • A graph of the above function, demonstrating that the limit at $x_0$ does not exist.
  • The limit as the function approaches $x_0$ from the left does not equal the limit as the function approaches $x_0$ from the right, so the limit of the function at $x_0$ does not exist.

• Precise Definition of a Limit

  • The $(\varepsilon,\delta)$-definition of limit (the "epsilon-delta definition") is a formalization of the notion of limit.
  • The $(\varepsilon,\delta)$-definition of limit ("epsilon-delta definition of limit") is a formalization of the notion of limit.
  • The $(\varepsilon,\delta)$-definition of limit is a formalization of the notion of limit.
  • Therefore, the limit of this function at infinity exists.
  • Therefore, the limit of this function at infinity exists.

• Calculating Limits Using the Limit Laws

  • Limits of functions can often be determined using simple laws, such as L'Hôpital's rule and squeeze theorem.
  • Limits of functions can often be determined using simple laws.
  • It is typically used to confirm the limit of a function via comparison with two other functions whose limits are known or easily computed.
  • Let $I$ be an interval having the point $a$ as a limit point.
  • Calculate a limit using simple laws, such as L'Hôpital's Rule or the squeeze theorem

• Infinite Limits

  • Limits involving infinity can be formally defined using a slight variation of the $(\varepsilon, \delta)$-definition.
  • Limits involving infinity can be formally defined using a slight variation of the $(\varepsilon, \delta)$-definition.
  • If the degree of $p$ is less than the degree of $q$, the limit is $0$.
  • If the limit at infinity exists, it represents a horizontal asymptote at $y = L$.
  • Therefore, the limit of this function at infinity exists.

• Finding Limits Algebraically

  • This set of rules is often called the algebraic limit theorem, expressed formally as follows:
  • In each case above, when the limits on the right do not exist (or, in the last case, when the limits in both the numerator and the denominator are zero), the limit on the left, called an indeterminate form, may nonetheless still exist—this depends on the functions f and g.
  • These rules are also valid for one-sided limits, for the case $p = \pm$, and also for infinite limits using the following rules:
  • The limit of $f(x)= \frac{-1}{(x+4)} + 4$ as $x$ goes to infinity can be segmented down into two parts: the limit of $\frac{−1}{(x+4)}$ and the limit of $4$.
  • Therefore, the limit of $f(x)$ as $x$ goes to infinity is $4$.

• Limits and Continuity

  • A study of limits and continuity in multivariable calculus yields counter-intuitive results not demonstrated by single-variable functions.
  • A study of limits and continuity in multivariable calculus yields many counter-intuitive results not demonstrated by single-variable functions .
  • For example, there are scalar functions of two variables with points in their domain which give a particular limit when approached along any arbitrary line, yet give a different limit when approached along a parabola.
  • However, when the origin is approached along a parabola $y = x^2$, it has a limit of $0.5$.
  • Since taking different paths toward the same point yields different values for the limit, the limit does not exist.

• Trigonometric Limits

  • This equation can be proven with the first limit and the trigonometric identity $1 - \cos^2 x = \sin^2 x$.

• Indeterminate Forms and L'Hôpital's Rule

  • Indeterminate forms like $\frac{0}{0}$ have no definite value; however, when a limit is indeterminate, l'Hôpital's rule can often be used to evaluate it.
  • In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression obtained in the context of limits.
  • Limits involving algebraic operations are often performed by replacing subexpressions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form.
  • More formally, the fact that the functions $f$ and $g$ both approach $0$ as $x$ approaches some limit point $c$ is not enough information to evaluate the limit $\lim_{x\to c}\frac{f(x)}{g(x)}$.
  • In calculus, l'Hôpital's rule uses derivatives to help evaluate limits involving indeterminate forms.

• Improper Integrals

  • But that conceals the limiting process.
  • Specifically, an improper integral is a limit of one of two forms.
  • First, an improper integral could be a limit of the form:
  • Second, an improper integral could be a limit of the form:
  • However, the improper integral does exist if understood as the limit