3.5 Limits at Infinity - Introduction to Calculus

Limits Approaching Positive or Negative Infinity

Sometimes, we will be interested in determining what happens as $x$ gets really big, either in the positive or negative direction.

Function

\displaystyle y=\frac{2x^2+3x+1}{x^2-10x+100}

Leading numerator coefficient2Direction Distance · power of ten1

\displaystyle y=\frac{2x^2+3x+1}{x^2-10x+100}

Input x	Function value
10	2.31000000
100	2.23087912
1000	2.02302899
10000	2.00230031
100000	2.00023000

Selected input: 10. Function value: 2.31000000.

In either direction the limit is 2, the ratio of leading quadratic coefficients.

Unlike a limit at a finite point, $x\to\infty$ or $x\to-\infty$ follows the function farther and farther along its domain. Always check that the real domain extends in the required direction.

QUESTION

Does

\displaystyle \displaystyle\lim_{x\to-\infty}\dfrac1{\sqrt[4]{x}}

exist over the real numbers?

Evaluate the following limits over the real numbers. Check the domain as well as the size of each expression.

(a) $\displaystyle \lim\limits_{x\to\infty}\frac1x$

(b) $\displaystyle \lim\limits_{x\to-\infty}\frac1x$

(d) $\displaystyle \lim\limits_{x\to-\infty}\dfrac{-238}{23x^{1/4}}$

Comparing the Highest Powers

For problems like $\lim\limits_{x\to\infty}\dfrac{2x^2+3x+1}{x^2-10x+100}$ , a good first step is to divide the top and the bottom by the highest power of $x$ appearing in the denominator.

Remember that $\sqrt{x^2}=|x|$ For $x>0$ this equals $x$ ; for $x<0$ it equals $-x$ This distinction matters when a square root appears in a limit at infinity.

Evaluate the following limits.

(a) $\displaystyle \lim\limits_{x\to\infty}\dfrac{2x^2+3x+1}{x^2-10x+100}$

(b) $\displaystyle \lim\limits_{x\to-\infty}\dfrac{3-2x^3}{1+x+x^2}$

(d) $\displaystyle \lim\limits_{t\to\infty}\left(\sqrt{3t^2+2t+5}+t\right)$

Additional comparison for (c): The source's worked solution instead starts with $\displaystyle \sqrt{x^2-x+1}-x$ Evaluate that difference too, and compare it with the expression in the question.

Limits Approaching Positive or Negative Infinity

Sometimes, we will be interested in determining what happens as

x

gets really big, either in the positive or negative direction.

Function

\displaystyle y=\frac{2x^2+3x+1}{x^2-10x+100}

Leading numerator coefficient2Direction Distance · power of ten1

\displaystyle y=\frac{2x^2+3x+1}{x^2-10x+100}

Input x	Function value
10	2.31000000
100	2.23087912
1000	2.02302899
10000	2.00230031
100000	2.00023000

Selected input: 10. Function value: 2.31000000.

In either direction the limit is 2, the ratio of leading quadratic coefficients.

Unlike a limit at a finite point,

x\to\infty

x\to-\infty

follows the function farther and farther along its domain. Always check that the real domain extends in the required direction.

QUESTION

Does

\displaystyle \displaystyle\lim_{x\to-\infty}\dfrac1{\sqrt[4]{x}}

exist over the real numbers?

Evaluate the following limits over the real numbers. Check the domain as well as the size of each expression.

(a) $\displaystyle \lim\limits_{x\to\infty}\frac1x$

(b) $\displaystyle \lim\limits_{x\to-\infty}\frac1x$

(d) $\displaystyle \lim\limits_{x\to-\infty}\dfrac{-238}{23x^{1/4}}$

Comparing the Highest Powers

For problems like

\lim\limits_{x\to\infty}\dfrac{2x^2+3x+1}{x^2-10x+100}

, a good first step is to divide the top and the bottom by the highest power of

x

appearing in the denominator.

Remember that

\sqrt{x^2}=|x|

For

x>0

this equals

x

; for

x<0

it equals

-x

This distinction matters when a square root appears in a limit at infinity.

Evaluate the following limits.

(a) $\displaystyle \lim\limits_{x\to\infty}\dfrac{2x^2+3x+1}{x^2-10x+100}$

(b) $\displaystyle \lim\limits_{x\to-\infty}\dfrac{3-2x^3}{1+x+x^2}$

(d) $\displaystyle \lim\limits_{t\to\infty}\left(\sqrt{3t^2+2t+5}+t\right)$

Additional comparison for (c): The source's worked solution instead starts with $\displaystyle \sqrt{x^2-x+1}-x$ Evaluate that difference too, and compare it with the expression in the question.

Limits at Infinity

Limits Approaching Positive or Negative Infinity

Follow a function toward infinity

KNOWLEDGE QUIZ

Check behavior and domain at infinity

Comparing the Highest Powers

Compare growth at infinity

Limits at Infinity

Limits Approaching Positive or Negative Infinity

Follow a function toward infinity

KNOWLEDGE QUIZ

Check behavior and domain at infinity

Comparing the Highest Powers

Compare growth at infinity