Improper Integrals

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.8 – Improper Integrals

Calculus I

Prerequisite Knowledge

This section contains mathematical content from previous courses that are necessary for improper integrals.

Limits

Definition: Limit

The limit of $f(x)$$f(x)$ as $x$$x$ approaches $a$$a$ is $L$$L$. This is written as:

${\displaystyle \underset{x\to a}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=L$

NOTE: Here is how to interpret the symbols in the definition of a limit.

• $a$$a$ is a Real-valued $x$$x$-value, i.e., $a\in \mathbb{R}$$a\in\mathbb{R}$.

  • $a$$a$ can be approached from the left, i.e., from smaller $x$$x$-values in a local neighborhood of (i.e., near) $a$$a$; this is written as ${\displaystyle \underset{x\to a^{-}}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow a^-}{f(x)}=L$
  • $a$$a$ can be approached from the right, i.e., from larger $x$$x$-values in a local neighborhood of (i.e., near) $a$$a$; this is written as ${\displaystyle \underset{x\to a^{+}}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow a^+}{f(x)}=L$
  • $a$$a$ can be approached simultaneously from the left and the right in a local neighborhood of (i.e., near) $a$$a$. this is written as ${\displaystyle \underset{x\to a}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=L$
  • NOTE: $\mathrm{\infty }$$\infty$ is not a Real number.

• $L$$L$ is a $y$$y$-value

  • $L$$L$ may be the $y$$y$-value of $f(x)$$f(x)$, i.e., $f(a)$$f(a)$;
  • $L$$L$ does not need to be the $y$$y$-value of $f(x)$$f(x)$;
  • $L$$L$ may not exist; in this case, we say “the limit does not exist” or “the limit DNE”
  • $L$$L$ may be a non-real number; in this case, we say “the limit DNE”.

Infinite Limits

• If ${\displaystyle \underset{x\to a}{lim}f(x)=\text{DNE}}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=\text{DNE}$, i.e., $f(x)$$f(x)$ becomes large without bound, then the limit does not exist. NOTE: DNE is an abbreviation of does not exist.
• If ${\displaystyle \underset{x\to a}{lim}f(x)=\text{DNE}}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=\text{DNE}$, i.e., $f(x)$$f(x)$ becomes small without bound, then the limit does not exist.

NOTE: $\mathrm{\infty }$$\infty$ is NOT a real number, i.e., $\mathrm{\infty }\notin \mathbb{R}$$\infty\notin\mathbb{R}$.

Limits at Infinity

Definition: Limit at Infinity

Let $f$$f$ be a function defined on some open interval $(a,\mathrm{\infty })$$(a,\infty)$. Then ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow\infty}{f(x)}=L$ means that as $x$$x$ becomes large without bound, the values of $f(x)$$f(x)$ become arbitrarily close to $L$$L$.

Definition: Limit at Negative Infinity

Let $f$$f$ be a function defined on some open interval $(-\mathrm{\infty },a)$$(-\infty,a)$. Then ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f(x)=L}$$\displaystyle\lim_{x\rightarrow -\infty}{f(x)}=L$ means that as $x$$x$ becomes small without bound, the values of $f(x)$$f(x)$ become arbitrarily close to $L$$L$.

NOTE: Since $\mathrm{\infty }$$\infty$ is NOT a real number, then $f(\mathrm{\infty })$$f(\infty)$ is NOT meaningful.

Evaluating Limits

Definition: Direct Substitution

• If $f(a)=L$$f(a)=L$, then ${\displaystyle \underset{x\to a}{lim}f(x)=f(a)=L}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=f(a)=L$.
• If $f(a)=L$$f(a)=L$, then ${\displaystyle \underset{x\to a^{-}}{lim}f(x)=f(a)=L}$$\displaystyle\lim_{x\rightarrow a^{-}}{f(x)}=f(a)=L$.
• If $f(a)=L$$f(a)=L$, then ${\displaystyle \underset{x\to a^{+}}{lim}f(x)=f(a)=L}$$\displaystyle\lim_{x\rightarrow a^{+}}{f(x)}=f(a)=L$.

Indeterminate Forms of Limits

Definition: Indeterminate Quotients

1. Type ${\displaystyle \frac{0}{0}}$$\displaystyle\frac{0}{0}$: If ${\displaystyle \underset{x\to a}{lim}f(x)=0}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=0$ and ${\displaystyle \underset{x\to a}{lim}g(x)=0}$$\displaystyle\lim_{x\rightarrow a}{g(x)}=0$, then ${\displaystyle \underset{x\to a}{lim}\frac{f(x)}{g(x)}}$$\displaystyle\lim_{x\rightarrow a}{\frac{f(x)}{g(x)}}$ may or may not exist.
2. Type ${\displaystyle \frac{\mathrm{\infty }}{\mathrm{\infty }}}$$\displaystyle\frac{\infty}{\infty}$: If ${\displaystyle \underset{x\to a}{lim}f(x)}$$\displaystyle\lim_{x\rightarrow a}{f(x)}$ becomes large without bound and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes large without bound, then ${\displaystyle \underset{x\to a}{lim}\frac{f(x)}{g(x)}}$$\displaystyle\lim_{x\rightarrow a}{\frac{f(x)}{g(x)}}$ may or may not exist.

Definition: Indeterminate Differences

3. Type ${\displaystyle \mathrm{\infty }-\mathrm{\infty }}$$\displaystyle\infty -\infty$: If ${\displaystyle \underset{x\to a}{lim}f(x)}$$\displaystyle\lim_{x\rightarrow a}{f(x)}$ becomes large without bound and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes large without bound, then ${\displaystyle \underset{x\to a}{lim}[f(x)-g(x)]}$$\displaystyle\lim_{x\rightarrow a}{\left[f(x)-g(x)\right]}$ may or may not exist.

Definition: Indeterminate Products

4. Type ${\displaystyle 0\cdot \mathrm{\infty }}$$\displaystyle 0\cdot\infty$: If ${\displaystyle \underset{x\to a}{lim}f(x)=0}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=0$ and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes large without bound, then ${\displaystyle \underset{x\to a}{lim}[f(x)g(x)]}$$\displaystyle\lim_{x\rightarrow a}{\left[f(x)g(x)\right]}$ may or may not exist.

Definition: Indeterminate Powers

5. Type ${\displaystyle 0^0}$$\displaystyle 0^0$: If ${\displaystyle \underset{x\to a}{lim}f(x)=0}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=0$ and ${\displaystyle \underset{x\to a}{lim}g(x)=0}$$\displaystyle\lim_{x\rightarrow a}{g(x)}=0$, then ${\displaystyle \underset{x\to a}{lim}f(x{)}^{g(x)}}$$\displaystyle\lim_{x\rightarrow a}{f(x)^{g(x)}}$ may or may not exist.
6. Type ${\displaystyle {\mathrm{\infty }}^0}$$\displaystyle \infty^0$: If ${\displaystyle \underset{x\to a}{lim}f(x)}$$\displaystyle\lim_{x\rightarrow a}{f(x)}$ becomes large without bound and ${\displaystyle \underset{x\to a}{lim}g(x)=0}$$\displaystyle\lim_{x\rightarrow a}{g(x)}=0$, then ${\displaystyle \underset{x\to a}{lim}f(x{)}^{g(x)}}$$\displaystyle\lim_{x\rightarrow a}{f(x)^{g(x)}}$ may or may not exist.
7. Type ${\displaystyle 1^{\mathrm{\infty }}}$$\displaystyle 1^{\infty}$: If ${\displaystyle \underset{x\to a}{lim}f(x)=1}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=1$ and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes large without bound, then ${\displaystyle \underset{x\to a}{lim}f(x{)}^{g(x)}}$$\displaystyle\lim_{x\rightarrow a}{f(x)^{g(x)}}$ may or may not exist.

l'Hôpital's Rule

NOTE: l'Hôpital's Rule is valid only for indeterminate quotients.

Definition: l'Hôpital's Rule

Suppose $f$$f$ and $g$$g$ are differentiable and $g^{\prime }(x)\ne 0$$g^{\,\prime}(x)\ne 0$ on some open interval that contains $a$$a$. Suppose that

• ${\displaystyle \underset{x\to a}{lim}f(x)=0}$$\displaystyle\lim_{x\rightarrow a}{f(x)}=0$ and ${\displaystyle \underset{x\to a}{lim}g(x)=0}$$\displaystyle\lim_{x\rightarrow a}{g(x)}=0$; or
• ${\displaystyle \underset{x\to a}{lim}f(x)}$$\displaystyle\lim_{x\rightarrow a}{f(x)}$ becomes large without bound and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes large without bound; or
• ${\displaystyle \underset{x\to a}{lim}f(x)}$$\displaystyle\lim_{x\rightarrow a}{f(x)}$ becomes small without bound and ${\displaystyle \underset{x\to a}{lim}g(x)}$$\displaystyle\lim_{x\rightarrow a}{g(x)}$ becomes small without bound

then ${\displaystyle \underset{x\to a}{lim}\frac{f(x)}{g(x)}=\underset{x\to a}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}}$$\displaystyle\lim_{x\rightarrow a}{\frac{f(x)}{g(x)}}=\lim_{x\rightarrow a}{\frac{f^{\,\prime}(x)}{g^{\,\prime}(x)}}$

Practice

Evaluate the following limits:

1. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\left[x{e}^{-x}\right]}$$\displaystyle \lim_{x\rightarrow\infty}\left[xe^{-x}\right]$
2. ${\displaystyle \underset{x\to 1^{+}}{lim}{e}^{x^2-x}}$$\displaystyle \lim_{x\rightarrow 1^{+}}e^{x^2-x}$
3. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x}{1+x^2}}$$\displaystyle \lim_{x\rightarrow\infty}\frac{x}{1+x^2}$
4. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{2x^3+5}{1+x^3}}$$\displaystyle \lim_{x\rightarrow\infty}\frac{2x^3+5}{1+x^3}$
5. ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{1+x^6}{1+x^3}}$$\displaystyle \lim_{x\rightarrow -\infty}\frac{1+x^6}{1+x^3}$
6. ${\displaystyle \underset{x\to 0}{lim}\frac{x}{1+x^2}}$$\displaystyle \lim_{x\rightarrow 0}\frac{x}{1+x^2}$
7. ${\displaystyle \underset{x\to 2^{-}}{lim}\frac{x^2-x-2}{x-2}}$$\displaystyle \lim_{x\rightarrow 2^{-}}\frac{x^2-x-2}{x-2}$
8. ${\displaystyle \underset{x\to 0}{lim}\frac{e^x-e^{-x}-2x}{x-\sin (x)}}$$\displaystyle \lim_{x\rightarrow 0}\frac{e^x-e^{-x}-2x}{x-\sin{(x)}}$
9. ${\displaystyle \underset{x\to 0}{lim}[x\ln (x)]}$$\displaystyle \lim_{x\rightarrow 0}\left[x\ln{(x)}\right]$
10. ${\displaystyle \underset{x\to 1}{lim}\frac{x^8-1}{x^5-1}}$$\displaystyle \lim_{x\rightarrow 1}{\frac{x^8-1}{x^5-1}}$
11. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{x}^3{e}^{-x^2}}$$\displaystyle \lim_{x\rightarrow \infty}{x^3 e^{-x^2}}$

Check Your Work

Use Technology to Verify Limits

NOTE: The use of technology in this activity is to verify, i.e., check, your answers. You must show work on assessments to receive credit for all Calculus I work.

Mathematica

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(* Example from Practice 01-1: Limit at infinity of xe^(-x) *)
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Limit[x*E^(-x), x->Infinity]
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(* Example from Practice 01-5: Limit at -infinity of (1+x^6)/(1+x^3) *)
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Limit[(1+x^6)/(1+x^3), x->-Infinity]
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(* Example from Practice 01-2: Limit to 1 from right of e^(x^2-x) *)
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Limit[E^(x^2-x), x->1, Direction->-1]
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(* OR: Limit to 1 from right of e^(x^2-x) *)
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Limit[E^(x^2-x), x->1, Direction->"FromAbove"]
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(* Example from Practice 01-7: Limit to 2 from left of (x^2-x-2)(x-2) *)
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Limit[(x^2-x-2)(x-2), x->2, Direction->1]
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(* OR: Limit to 2 from left of (x^2-x-2)(x-2) *)
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Limit[(x^2-x-2)(x-2), x->2, Direction->"FromBelow"]
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(* Example from Practice 01-10: Limit to 1 of (x^8-1)(x^5-1) *)
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Limit[(x^8-1)(x^5-1), x->1]

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. Limit[ ] has two arguments:

   1. The function in the argument of the limit;

   2. $x\text{->}a$$x\text{->}a$, how the limit is to be determined.

   3. The direction:

      1. $\text{Direction->"FromAbove"}$$\text{Direction->"FromAbove"}$ for limits from the right;
      2. $\text{Direction->"FromAbove"}$$\text{Direction->"FromAbove"}$ for limits from the left;
      3. Do not use Direction for limits from both sides.

3. Mathematica uses $\text{Sqrt[}\text{]}$$\text{Sqrt[\,\,]}$ for $\sqrt{}$$\sqrt{\ \ \ }$; $\text{Cos[ ]}$$\text{Cos[\ \ ]}$ for $\cos (x)$$\cos{(x)}$; $\text{Log[ ]}$$\text{Log[\ \ ]}$ for $\ln (x)$$\ln{(x)}$; $\text{E}\wedge x$$\text{E}\!\wedge\!x$ for $e^x$$e^x$; etc.

4. Remember, correct Mathematica code will be all black except for variables.

5. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator.

6. If Mathematica returns either $\mathrm{\infty }$$\infty$ or $-\mathrm{\infty }$$-\infty$, then the limit DNE.

7. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

NOTE: Python is used here to verify limits at infinity. On all Assessments, you must show that you understand the Calculus I concept of evaluating limits at infinity.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.

2. All code is included above with comment codes that explain the executable code.

3. limit( ) command has three arguments:

   1. The function, $f$$f$, in the argument of the limit;
   2. $x$$x$​, where $x$$x$​​ is the variable used in the function.
   3. $a$$a$, where $a$$a$ is written as $\text{oo}$$\text{oo}$ for $\mathrm{\infty }$$\infty$ or $\text{-oo}$$\text{-oo}$ for $-\mathrm{\infty }$$-\infty$. [$\text{oo}$$\text{oo}$ is two lower-case ‘oo’]

4. Python uses $\text{sqrt(}\text{)}$$\text{sqrt(\,\,)}$ for $\sqrt{}$$\sqrt{\ \ \ }$; $\text{log( )}$$\text{log(\ \ )}$ for $\ln (x)$$\ln{(x)}$; $\text{exp(x)}$$\text{exp(x)}$ for $e^x$$e^x$; etc.

5. You only need to change:

   1. the function in line 9 of the code;
   2. $a$$a$ in line 12 of the code.

6. The code will output all three directional limits. Choose the appropriate one.

7. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator.

8. Python requires $\ast$$*$ for explicit multiplication; $\ast \ast$$**$​ for exponents; $\text{exp(x)}$$\text{exp(x)}$ for $e^x$$e^x$; etc.

9. Remember to use parens $($$\left(\right.$ and $)$$\left.\right)$​ to group multiple terms in the numerator or denominator of a rational expression.

10. If Python returns either $\mathrm{\infty }$$\infty$ or $-\mathrm{\infty }$$-\infty$, then the limit DNE.

11. To execute code, press the “View the result” button:

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Tuesday, 15 September 2020 12:39 EDT Last Modified: Monday, 14 February 2022 - 07:01 (EST)