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What Is The Behavior Of Limits At Infinity?

Published Aug 29, 2025 8 min read
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The behavior of limits at infinity describes the "end behavior" of a function, revealing what happens to the function's output ( yy 𝑦 -values) as the input (xx

π‘₯

) grows without bound, either positively (xβ†’βˆžx right arrow infinity

π‘₯β†’βˆž

) or negatively (xβ†’βˆ’βˆžx right arrow negative infinity

π‘₯β†’βˆ’βˆž

). The analysis of limits at infinity is fundamental to determining if a function has a horizontal asymptote.

The core concept of limits at infinity

As xx

π‘₯

approaches infinity or negative infinity, the function's behavior can follow one of three general paths:

  • Approach a finite number: The function's value gets arbitrarily close to a specific, finite number Lcap L

    𝐿

    . If limxβ†’βˆžf(x)=Llimit over x right arrow infinity of f of x equals cap L

    limπ‘₯β†’βˆžπ‘“(π‘₯)=𝐿

    , the line y=Ly equals cap L

    𝑦=𝐿

    is a horizontal asymptote.

  • Approach infinity or negative infinity: The function's value grows or shrinks without bound, tending toward ∞infinity

    ∞

    or βˆ’βˆžnegative infinity

    βˆ’βˆž

    .

  • Oscillate: The function's value bounces between different numbers, never settling on a single value, so the limit does not exist.

Evaluating limits at infinity for different function types

Rational functions

For rational functionsβ€”a ratio of two polynomials, f(x)=P(x)Q(x)f of x equals the fraction with numerator cap P open paren x close paren and denominator cap Q open paren x close paren end-fraction

𝑓(π‘₯)=𝑃(π‘₯)𝑄(π‘₯)

β€”the limit at infinity is determined by comparing the degrees of the numerator, P(x)cap P open paren x close paren

𝑃(π‘₯)

, and the denominator, Q(x)cap Q open paren x close paren

𝑄(π‘₯)

. The dominant term in each polynomial (the term with the highest power of xx

π‘₯

) dictates the function's end behavior. A reliable algebraic technique is to divide every term in the function by the highest power of xx

π‘₯

in the denominator.

Let Ncap N

𝑁

be the degree of the numerator and Dcap D

𝐷

be the degree of the denominator.

  • **If N<Dcap N is less than cap D

    𝑁<𝐷

    :** The denominator grows much faster than the numerator. The limit is 0, and there is a horizontal asymptote at y=0y equals 0

    𝑦=0

    .

    • **Example:**limxβ†’βˆžx2βˆ’1x3+2=limxβ†’βˆžx2x3βˆ’1x3x3x3+2x3=limxβ†’βˆž1xβˆ’1x31+2x3=0βˆ’01+0=0limit over x right arrow infinity of the fraction with numerator x squared minus 1 and denominator x cubed plus 2 end-fraction equals limit over x right arrow infinity of the fraction with numerator the fraction with numerator x squared and denominator x cubed end-fraction minus the fraction with numerator 1 and denominator x cubed end-fraction and denominator the fraction with numerator x cubed and denominator x cubed end-fraction plus the fraction with numerator 2 and denominator x cubed end-fraction end-fraction equals limit over x right arrow infinity of the fraction with numerator 1 over x end-fraction minus the fraction with numerator 1 and denominator x cubed end-fraction and denominator 1 plus the fraction with numerator 2 and denominator x cubed end-fraction end-fraction equals the fraction with numerator 0 minus 0 and denominator 1 plus 0 end-fraction equals 0

      limπ‘₯β†’βˆžπ‘₯2βˆ’1π‘₯3+2=limπ‘₯β†’βˆžπ‘₯2π‘₯3βˆ’1π‘₯3π‘₯3π‘₯3+2π‘₯3=limπ‘₯β†’βˆž1π‘₯βˆ’1π‘₯31+2π‘₯3=0βˆ’01+0=0

      .

  • **If N=Dcap N equals cap D

    𝑁=𝐷

    :** The numerator and denominator grow at the same rate. The limit is the ratio of the leading coefficients.

    • **Example:**limxβ†’βˆž5x2+32x2+4x=limxβ†’βˆž5x2x2+3x22x2x2+4xx2=limxβ†’βˆž5+3x22+4x=5+02+0=52limit over x right arrow infinity of the fraction with numerator 5 x squared plus 3 and denominator 2 x squared plus 4 x end-fraction equals limit over x right arrow infinity of the fraction with numerator the fraction with numerator 5 x squared and denominator x squared end-fraction plus the fraction with numerator 3 and denominator x squared end-fraction and denominator the fraction with numerator 2 x squared and denominator x squared end-fraction plus the fraction with numerator 4 x and denominator x squared end-fraction end-fraction equals limit over x right arrow infinity of the fraction with numerator 5 plus the fraction with numerator 3 and denominator x squared end-fraction and denominator 2 plus 4 over x end-fraction end-fraction equals the fraction with numerator 5 plus 0 and denominator 2 plus 0 end-fraction equals five-halves

      limπ‘₯β†’βˆž5π‘₯2+32π‘₯2+4π‘₯=limπ‘₯β†’βˆž5π‘₯2π‘₯2+3π‘₯22π‘₯2π‘₯2+4π‘₯π‘₯2=limπ‘₯β†’βˆž5+3π‘₯22+4π‘₯=5+02+0=52

      .

  • **If N>Dcap N is greater than cap D

    𝑁>𝐷

    :** The numerator grows faster than the denominator. The limit is **∞infinity

    ∞

    or βˆ’βˆžnegative infinity

    βˆ’βˆž** , and there is no horizontal asymptote. The sign is determined by the signs of the leading coefficients.

    • **Example:**limxβ†’βˆž4x3xβˆ’1=limxβ†’βˆž4x3xxxβˆ’1x=limxβ†’βˆž4x21βˆ’1x=∞1=∞limit over x right arrow infinity of the fraction with numerator 4 x cubed and denominator x minus 1 end-fraction equals limit over x right arrow infinity of the fraction with numerator the fraction with numerator 4 x cubed and denominator x end-fraction and denominator x over x end-fraction minus 1 over x end-fraction end-fraction equals limit over x right arrow infinity of the fraction with numerator 4 x squared and denominator 1 minus 1 over x end-fraction end-fraction equals the fraction with numerator infinity and denominator 1 end-fraction equals infinity

      limπ‘₯β†’βˆž4π‘₯3π‘₯βˆ’1=limπ‘₯β†’βˆž4π‘₯3π‘₯π‘₯π‘₯βˆ’1π‘₯=limπ‘₯β†’βˆž4π‘₯21βˆ’1π‘₯=∞1=∞

      .

Functions involving radicals

For limits at infinity involving square roots, particularly in rational expressions, the highest power of xx

π‘₯

in the denominator must be carefully identified. When xβ†’βˆ’βˆžx right arrow negative infinity

π‘₯β†’βˆ’βˆž

, the term x2the square root of x squared end-root

π‘₯2√

should be treated as βˆ’xnegative x

βˆ’π‘₯

, not xx

π‘₯

, to account for the negativity of xx

π‘₯

.

  • **Example:**limxβ†’βˆ’βˆžx2+12xβˆ’5limit over x right arrow negative infinity of the fraction with numerator the square root of x squared plus 1 end-root and denominator 2 x minus 5 end-fraction

    limπ‘₯β†’βˆ’βˆžπ‘₯2+1√2π‘₯βˆ’5

    • Divide the numerator and denominator by xx

      π‘₯

      : limxβ†’βˆ’βˆžx2+1x2xβˆ’5x=limxβ†’βˆ’βˆžβˆ’x2+1x22βˆ’5x=limxβ†’βˆ’βˆžβˆ’1+1x22βˆ’5x=βˆ’1+02βˆ’0=βˆ’12limit over x right arrow negative infinity of the fraction with numerator the fraction with numerator the square root of x squared plus 1 end-root and denominator x end-fraction and denominator the fraction with numerator 2 x minus 5 and denominator x end-fraction end-fraction equals limit over x right arrow negative infinity of the fraction with numerator negative the square root of the fraction with numerator x squared plus 1 and denominator x squared end-fraction end-root and denominator 2 minus 5 over x end-fraction end-fraction equals limit over x right arrow negative infinity of the fraction with numerator negative the square root of 1 plus the fraction with numerator 1 and denominator x squared end-fraction end-root and denominator 2 minus 5 over x end-fraction end-fraction equals the fraction with numerator negative the square root of 1 plus 0 end-root and denominator 2 minus 0 end-fraction equals negative one-half

      limπ‘₯β†’βˆ’βˆžπ‘₯2+1√π‘₯2π‘₯βˆ’5π‘₯=limπ‘₯β†’βˆ’βˆžβˆ’π‘₯2+1π‘₯22βˆ’5π‘₯=limπ‘₯β†’βˆ’βˆžβˆ’1+1π‘₯22βˆ’5π‘₯=βˆ’1+0√2βˆ’0=βˆ’12

      .

Exponential and logarithmic functions

The behavior of these functions at infinity is dictated by their intrinsic growth rates.

  • **Exponential functions (f(x)=bxf of x equals b to the x-th power

    𝑓(π‘₯)=𝑏π‘₯

    ):**

    • **b>1b is greater than 1

      𝑏>1

      :**limxβ†’βˆžbx=∞limit over x right arrow infinity of b to the x-th power equals infinity

      limπ‘₯β†’βˆžπ‘π‘₯=∞

      and limxβ†’βˆ’βˆžbx=0limit over x right arrow negative infinity of b to the x-th power equals 0

      limπ‘₯β†’βˆ’βˆžπ‘π‘₯=0

      .

    • **0<b<10 is less than b is less than 1

      0<𝑏<1

      :**limxβ†’βˆžbx=0limit over x right arrow infinity of b to the x-th power equals 0

      limπ‘₯β†’βˆžπ‘π‘₯=0

      and limxβ†’βˆ’βˆžbx=∞limit over x right arrow negative infinity of b to the x-th power equals infinity

      limπ‘₯β†’βˆ’βˆžπ‘π‘₯=∞

      .

  • **Logarithmic functions (f(x)=logb(x)f of x equals log base b of x

    𝑓(π‘₯)=log𝑏(π‘₯)

    ):**

    • **b>1b is greater than 1

      𝑏>1

      :**limxβ†’βˆžlogb(x)=∞limit over x right arrow infinity of log base b of x equals infinity

      limπ‘₯β†’βˆžlog𝑏(π‘₯)=∞

      . The domain of logb(x)log base b of x

      log𝑏(π‘₯)

      is only positive numbers, so limxβ†’βˆ’βˆžlimit over x right arrow negative infinity of

      limπ‘₯β†’βˆ’βˆž

      is not considered.

Trigonometric functions

The standard trigonometric functions, such as sin(x)sine x

sin(π‘₯)

and cos(x)cosine x

cos(π‘₯)

, have no limit as xβ†’βˆžx right arrow infinity

π‘₯β†’βˆž

or xβ†’βˆ’βˆžx right arrow negative infinity

π‘₯β†’βˆ’βˆž

because they oscillate indefinitely between -1 and 1.

However, the Squeeze Theorem can be used for more complex expressions involving trigonometric functions.

  • **Example:**limxβ†’βˆžsin(x)xlimit over x right arrow infinity of sine x over x end-fraction

    limπ‘₯β†’βˆžsin(π‘₯)π‘₯

    • We know that -1≀sin(x)≀1negative 1 is less than or equal to sine x is less than or equal to 1

      βˆ’1≀sin(π‘₯)≀1

      .

    • Divide all parts of the inequality by xx

      π‘₯

      (for x>0x is greater than 0

      π‘₯>0

      ): -1x≀sin(x)x≀1xnegative 1 over x end-fraction is less than or equal to sine x over x end-fraction is less than or equal to 1 over x end-fraction

      βˆ’1π‘₯≀sin(π‘₯)π‘₯≀1π‘₯

      .

    • Since limxβ†’βˆž-1x=0limit over x right arrow infinity of negative 1 over x end-fraction equals 0

      limπ‘₯β†’βˆžβˆ’1π‘₯=0

      and limxβ†’βˆž1x=0limit over x right arrow infinity of 1 over x end-fraction equals 0

      limπ‘₯β†’βˆž1π‘₯=0

      , by the Squeeze Theorem, limxβ†’βˆžsin(x)x=0limit over x right arrow infinity of sine x over x end-fraction equals 0

      limπ‘₯β†’βˆžsin(π‘₯)π‘₯=0

      .

Connection to horizontal asymptotes

A function has a horizontal asymptote at y=Ly equals cap L

𝑦=𝐿

if either limxβ†’βˆžf(x)=Llimit over x right arrow infinity of f of x equals cap L

limπ‘₯β†’βˆžπ‘“(π‘₯)=𝐿

or limxβ†’βˆ’βˆžf(x)=Llimit over x right arrow negative infinity of f of x equals cap L

limπ‘₯β†’βˆ’βˆžπ‘“(π‘₯)=𝐿

.

It is important to evaluate the limit at both positive and negative infinity, as a function can have different horizontal asymptotes in each direction. For example, a rational function with a square root may have different limits as xβ†’βˆžx right arrow infinity

π‘₯β†’βˆž

and xβ†’βˆ’βˆžx right arrow negative infinity

π‘₯β†’βˆ’βˆž

.

Summary of key behaviors

Function Type Limit as xβ†’βˆžx right arrow infinity π‘₯β†’βˆž Limit as xβ†’βˆ’βˆžx right arrow negative infinity π‘₯β†’βˆ’βˆž
Rational (N<Dcap N is less than cap D 𝑁<𝐷 ) 0 0
Rational (N=Dcap N equals cap D 𝑁=𝐷 ) Ratio of leading coefficients Ratio of leading coefficients
Rational (N>Dcap N is greater than cap D 𝑁>𝐷 ) ±∞plus or minus infinity ±∞ ±∞plus or minus infinity ±∞
Polynomial (p(x)p open paren x close paren 𝑝(π‘₯) ) ±∞plus or minus infinity ±∞ (depends on leading term) ±∞plus or minus infinity ±∞ (depends on leading term)
Exponential (bx,b>1b to the x-th power comma b is greater than 1 𝑏π‘₯,𝑏>1 ) ∞infinity ∞ 0
Exponential (bx,0<b<1b to the x-th power comma 0 is less than b is less than 1 𝑏π‘₯,0<𝑏<1 ) 0 ∞infinity ∞
Logarithmic (logb(x),b>1log base b of x comma b is greater than 1 log𝑏(π‘₯),𝑏>1 ) ∞infinity ∞ Not applicable
Sine/Cosine Does not exist (oscillates) Does not exist (oscillates)
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