Limit

Limit[f,xx*]

gives the limit xx*f(x).

Limit[f,{x1,,xn}]

gives the nested limit f (x1,,xn).

Limit[f,{x1,,xn}{,,}]

gives the multivariate limit f (x1,,xn).

Details and Options

  • Limit is also known as function limit, directed limit, iterated limit, nested limit and multivariate limit.
  • Limit computes the limiting value f* of a function f as its variables x or xi get arbitrarily close to their limiting point x* or .
  • By using the character , entered as lim or \[Limit], with underscripts or subscripts, limits can be entered as follows:
  • flimit in the default direction
    flimit from above
    flimit from below
    flimit in the complex plane
    fLimit[f,{x1,,xn}]
  • For a finite limit point x* and {,,} and finite limit value f*:
  • Limit[f,xx*]f*for every there is such that 0<TemplateBox[{{x, -, {x, ^, *}}}, Abs]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, x, )}, -, {f, ^, *}}}, Abs]<epsilon
    Limit[f,{x1,,xn}{,,}]f*for every there is such that 0<TemplateBox[{{{, {{{x, _, 1}, -, {x, _, {(, 1, )}, ^, *}}, ,, ..., ,, {{x, _, n}, -, {x, _, {(, n, )}, ^, *}}}, }}}, Norm]<delta(epsilon,x^*) implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • For an infinite limit point and finite limit value f*:
  • Limit[f,x]f*for every there is such that implies TemplateBox[{{{f, (, x, )}, -, {f, ^, *}}}, Abs]<epsilon
    Limit[f,{x1,,xn}{,,}]f*for every there is such that implies TemplateBox[{{{f, (, {{x, _, 1}, ,, ..., ,, {x, _, n}}, )}, -, {f, ^, *}}}, Abs]<epsilon
  • Limit returns Indeterminate when it can prove the limit does not exist. MinLimit and MaxLimit can frequently be used to compute the minimum and maximum limit of a function if its limit does not exist.
  • Limit returns unevaluated or an Interval when no limit can be found. If an Interval is returned, there are no guarantees that this is the smallest possible interval.
  • The following options can be given:
  • Assumptions$Assumptionsassumptions on parameters
    DirectionRealsdirections to approach the limit point
    GenerateConditionsAutomaticwhether to generate conditions on parameters
    MethodAutomaticmethod to use
    PerformanceGoal"Quality"aspects of performance to optimize
  • Possible settings for Direction include:
  • Reals or "TwoSided"from both real directions
    "FromAbove" or -1from above or larger values
    "FromBelow" or +1from below or smaller values
    Complexesfrom all complex directions
    Exp[ θ]in the direction
    {dir1,,dirn}use direction diri for variable xi independently
  • DirectionExp[ θ] at x* indicates the direction tangent of a curve approaching the limit point x*.
  • Possible settings for GenerateConditions include:
  • Automaticnon-generic conditions only
    Trueall conditions
    Falseno conditions
    Nonereturn unevaluated if conditions are needed
  • Possible settings for PerformanceGoal include $PerformanceGoal, "Quality" and "Speed". With the "Quality" setting, Limit typically solves more problems or produces simpler results, but it potentially uses more time and memory.

Examples

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Basic Examples  (3)

Limit at a point of discontinuity:

Limit at infinity:

Limit from above:

Limit from below:

The two-sided limit does not exist:

Scope  (35)

Basic Uses  (5)

Find the limit at a point:

Find the limit at a symbolic point:

Find the limit at -Infinity:

The nested limit as first and then :

The nested limit as and then :

Compute the multivariate limit as :

Typeset Limits  (4)

Use lim to enter the character, and to create an underscript:

Take a limit from above or below by using a superscript or on the limit point:

After typing zero, use to create a superscript:

To specify a direction of Reals or Complexes, enter the domain as an underscript on the character:

Enter the rule as ->, use to create an underscript, and type reals to enter :

TraditionalForm formatting:

Elementary Functions  (6)

Polynomial and rational functions:

Algebraic functions:

Trigonometric functions:

A trigonometric function with a vertical asymptote:

A wildly oscillatory function with no limit at the origin:

Exponential functions:

Logarithmic functions:

Functions like Sqrt and Log have a two-sided limit along the negative reals:

If approached from above in the complex plane, the same limit value is reached:

However, approaching from below in the complex plane produces a different limit value:

This is due to branch cut where the imaginary part reverses sign as the axis is crossed:

The limit in the complex plane does not exist:

Elementary Functions at Infinity  (4)

Limits of algebraic functions at ±Infinity:

Limits of trigonometric functions at ±Infinity:

Limits of exponential and logarithmic functions at infinity:

Compute nested exponential-logarithmic limits:

Piecewise Functions  (5)

A discontinuous piecewise function:

A left-continuous piecewise function:

UnitStep is effectively a right-continuous piecewise function:

RealSign is effectively a discontinuous piecewise function:

Find the limit of Floor as x approaches from larger numbers:

Find the limit of Floor as x approaches from smaller numbers:

Special Functions  (4)

Limits involving Gamma:

Limits involving Bessel-type functions:

Limits involving exponential integrals:

At every non-positive even integer, Gamma approaches from the left and from the right:

The signs are reversed at the negative odd integers:

Nested Limits  (3)

Compute the nested limit as first and then :

The same result is obtained by computing two Limit expressions:

Computing the limit as first and then yields a different answer:

This is again equivalent to two nested limits:

The nested limit as first and then is :

The nested limit as first and then is :

Consider the function for two variables at the origin:

The iterated limit as and then is :

The iterated limit as and then is :

Since the value of the limit depends on the order, the bivariate limit does not exist:

Visualize the function and the values along the two axes computed previously:

Multivariate Limits  (4)

Compute the bivariate limit of a function as :

The limit value is if for all , there is a where implies epsilon>TemplateBox[{{{f, (, {x, ,, y}, )}, -, L}}, Abs]:

For this function, the value will suffice:

The function lies in between the two cones with slope :

Find the limit of a multivariate function:

Various iterated limits exist:

Approaching the origin along the curve yields a third result:

The true two-dimensional limit of the function does not exist:

Visualize the limit near the origin:

Find the limit of a bivariate function at the origin:

The true two-dimensional limit at the origin is zero:

Re-express the function in terms of polar coordinates:

The polar expression is bounded, and the limit as is:

Compute the limit of a trivariate function:

The limit at the origin does not exist:

Limits in the and planes do exist:

But the limit in the plane is direction dependent:

Visualize the function:

Options  (13)

Assumptions  (2)

Specify conditions on parameters using Assumptions:

Different assumptions can produce different results:

Parameterdependent limit:

Direction  (5)

Limit from below:

Equivalently:

Limit from above:

Equivalently:

Limits at piecewise discontinuities:

Limits at a simple pole:

Limits at a branch cut:

Compute the bivariate limit approach from different quadrants:

Approaching the origin from the first quadrant:

Equivalently:

Approaching the origin from the second quadrant:

Approaching the origin from the left half-plane:

Approaching the origin from the bottom half-plane:

Visualize the function:

GenerateConditions  (3)

Return a result without stating conditions:

This result is only valid if n>0:

Return unevaluated if the results depend on the value of parameters:

By default, conditions are generated that return a unique result:

By default, conditions are not generated if only special values invalidate the result:

With GenerateConditions->True, even these non-generic conditions are reported:

Method  (2)

Use Method{"AllowIndeterminateOutput"False} to avoid Indeterminate results:

For oscillatory functions, bounds will be returned as Interval objects:

Use Method{"AllowIntervalOutput"False} to avoid Interval object results:

PerformanceGoal  (1)

Use PerformanceGoal to avoid potentially expensive computations:

The default setting uses all available techniques to try to produce a result:

Applications  (23)

Geometry of Limits  (5)

The limit of the function as is :

This means that for values of close to , has a value close to :

Visualize the function:

The limit makes no statement about the value of at , which in this case is indeterminate:

The function does not have a limit as approaches :

The function has the limit zero:

In increasingly small regions around , continually bounces between , but gets increasingly flat:

The following rational function has a finite limit as :

Compute the that ensures that TemplateBox[{{{r, (, x, )}, -, 2}}, RealAbs]<epsilon whenever 0<TemplateBox[{{x, -, 1}}, RealAbs]<delta:

The complicated result can be simplified by focusing on in the range between 0 and 2:

The plot of always "leaves" the rectangle of height and width centered through the sides, not the top or bottom:

Find vertical and horizontal asymptotes of a rational function:

Compute where the denominator is zero:

Verify that the function approaches at the computed values:

Compute the limit value at :

Visualize the function and its asymptotes:

Find the non-vertical, linear asymptote of a function:

Slope of the asymptote:

Compute the asymptote's vertical intercept:

Visualize the function and its asymptote:

Discontinuities  (5)

Classify the continuity or discontinuity of f at the origin:

It is not defined at 0, so it cannot be continuous:

Moreover, the limit as x0 does not exist:

However, the limit from below exists:

The limit from above also exists, but has a different value:

Therefore, f has a jump discontinuity at 0:

Classify the continuity or discontinuity of f at the origin:

The function is defined:

The two-sided limit exists but does not equal the function value, so this is a removable discontinuity:

Find and classify the discontinuities of a piecewise function:

The function is not defined at zero so it cannot be continuous there:

The function tends to Infinity (on both sides), so this is an infinite discontinuity:

Next find where the limit does not equal the function:

The limit does exist at x==3, so this is a removable discontinuity:

The function is discontinuous at every multiple of :

For example, at the origin it gives rise to the indeterminate form :

At every even multiple of , the two-sided limit of exists:

This is also true at the odd multiples of , with a different limit:

However, at the half-integer multiples of , the two sided limit does not exist:

The function agrees with where both are defined, but it is also continuous at multiples of :

Show that is continuous at the origin:

Visualize and :

Determine whether the following function is continuous at the origin, and whether limits along rays exist:

The bivariate limit does not exist, so the function is not continuous:

The limit in the left half-plane exists and is zero, so any ray approaching from there has the same limit:

Approaching along the line with gives a result in terms of the slope:

Visualize the four rays with slope :

Derivatives  (5)

Compute the derivative of using the definition of derivative:

First compute the difference quotient:

The derivative is the limit as of the difference quotient:

Compute the derivative of at :

The limit of the difference quotient does not exist, so is not differentiable at the origin:

Note that the left and right limits of the difference quotient exist but are unequal:

In this case, the left and right derivatives equal the limits of from the left and right:

Visualize and its derivative; the former has a "kink" at zero, the latter a jump discontinuity:

Determine the differentiability of at :

The limit of the difference quotient exists, so is differentiable and :

Note that the limit as of does not exist, so is discontinuous:

Determine the differentiability of at the point :

The partial derivative with respect to x exists:

As does the one with respect to y:

However, the linearization condition lim_(r->r_0)(TemplateBox[{{{f, (, {x, ,, y}, )}, -, {f, (, {{x, _, 0}, ,, {y, _, 0}}, )}, -, {{p, (, {{x, _, 0}, ,, {y, _, 0}}, )}, ., {(, {r, -, {r, _, 0}}, )}}}}, RealAbs])/(TemplateBox[{{r, -, {r, _, 0}}}, Norm])=0 fails, so is not differentiable:

Visualize the function:

Note that the partial derivatives of exist everywhere:

But they are discontinuous at the point :

The derivative is defined as the limit of the difference quotient:

The second derivative can be computed by taking the limit of the second-order difference quotient:

Directly compute the mixed partial derivative by taking a limit:

Mathematical Constants and Expressions  (4)

Compute as a limit at Infinity:

The complementary limit at the origin:

Compute as a limit of Gamma functions:

Compute EulerGamma as a limit involving the Zeta function:

Compute EulerGamma as a limit of exponential integrals:

Other Applications  (4)

A function is said to be "little-o of " at , written , if lim_(x->a) TemplateBox[{{{(, {f, (, x, )}, )}, /, {(, {g, (, x, )}, )}}}, Abs]=0:

Similarly, is said to be "little-omega of ", written , if lim_(x->a) TemplateBox[{{{(, {f, (, x, )}, )}, /, {(, {g, (, x, )}, )}}}, Abs]=infty:

If , then :

It is possible for two functions to share neither relationship:

Moreover, neither relationship even holds between a function and itself:

Hence, and define partial orders on the functions:

if goes to zero faster than :

denotes the opposite relationship:

Note that the two lists are not exactly reversed, because and are incomparable:

From Taylor's theorem, if has continuous derivatives around , then :

This is the fifth-order Taylor polynomial at :

The definition of is that lim_(x->a) TemplateBox[{{{(, {f, (, x, )}, )}, /, {(, {g, (, x, )}, )}}}, Abs]=0:

Verify that :

In special relativity, the kinetic energy of a particle of mass and speed is given by:

The classical formula for kinetic energy is:

In the limit that the speed approaches zero, these two formulas agree:

Improper integrals:

Properties & Relations  (13)

Multiplicative constants can be moved outside a limit:

If f and g have finite limits, Limit is distributive over their sum:

If f and g have finite limits, Limit is distributive over their product:

Powers can be moved outside a limit:

Function composition and sequence limit operations can be interchanged for continuous functions:

This need not hold for discontinuous functions:

The "squeezing" or "sandwich" theorem:

This function is bounded by +/-TemplateBox[{x}, RealAbs]:

The limit of the bounding functions is zero, which proves the original limit was zero:

The squeezing theorem for limits at infinity:

This function is bounded by on the positive real axis:

The limit of the bounding functions is zero, which proves the original limit was zero:

Assumptions applies to parameters in the limit expression:

Direction places conditions on the limit variable:

Derivatives are defined in terms of limits:

The limit of a ratio can often be computed using L'Hôpital's rule:

Computing the ratio directly gives an indeterminate form 0/0:

The limit of the ratio equals the limit of the ratio of the derivatives:

In this case, f' and g' are continuous and can be computed via evaluation:

If Limit exists, then so does DiscreteLimit, and they have the same value:

The converse need not hold:

If Limit exists, then so does MaxLimit, and it has the same value:

If Limit exists, then so does MinLimit, and it has the same value:

Possible Issues  (1)

Limit may return an incorrect answer for an inexact input:

The result is correct when an exact input is used:

Numerical cancellations are behind the incorrect result:

Interactive Examples  (1)

In a sector bounded by a diameter and perpendicular chord, find the fraction occupied by the triangle:

If the disk has radius r, the area of the light blue shaded right triangle is:

Similarly, the total shaded area is the area of the whole sector minus the area of the white right triangle:

Compute the limit as ϕ approaches 0:

Neat Examples  (2)

Differentiation by integration:

Visualize a set of limits:

Introduced in 1988
 (1.0)
 |
Updated in 1996
 (3.0)
2003
 (5.0)
2017
 (11.2)