Gilbert Strang’s Calculus: Limits and Continuous Functions

Calculus

Limits

Continuity

Indeterminate Forms

L'Hospital Rule

Gilbert Strang

Author

Chao Ma

Published

February 28, 2026

Limit Definition

Limits are intuitive (“approach a value”), but calculus needs a strict definition.

For a sequence $a_n$, write \[ \lim_{n\to\infty} a_n = A \] if for every $\epsilon>0$, there exists an integer $N$ such that for all $n>N$, \[ |a_n-A|<\epsilon. \]

This is the $\epsilon$-$N$ definition: eventually, all terms stay within any target tolerance around $A$.

Epsilon neighborhood for sequence limits

Properties of Limit

If $\lim_{n\to\infty} a_n=A$ and $\lim_{n\to\infty} b_n=B$, then:

$a_n \pm b_n \to A \pm B$
$a_n b_n \to AB$
$a_n / b_n \to A/B$ when $B\neq 0$
$(a_n)^{b_n} \to A^B$ in regular cases

Dangerous!

For these properties, special endpoint cases become indeterminate forms:

In subtraction, if both limits are $\infty$, then $\infty-\infty$ is indeterminate.
In multiplication, if limits are $0$ and $\infty$, then $0\cdot\infty$ is indeterminate.
In division:
- if both limits are $0$, then $0/0$ is indeterminate.
- if both limits are $\infty$, then $\infty/\infty$ is indeterminate.
In powers:
- if base and exponent limits are both $0$, then $0^0$ is indeterminate.
- if base tends to $1$ and exponent tends to $\infty$, then $1^\infty$ is indeterminate.
- if base tends to $\infty$ and exponent tends to $0$, then $\infty^0$ is indeterminate.

Here are the 7 indeterminate forms (grouped as in your note):

Quotients (Fractions)
- $0/0$ (Numerator pulls to 0; Denominator pulls to $\infty$)
- $\infty/\infty$ (Numerator pulls to $\infty$; Denominator pulls to 0)
Products (Multiplication)
- $0\cdot\infty$ (One factor pulls to 0; the other pulls to $\infty$)
Differences (Subtraction)
- $\infty-\infty$ (Both terms pull the result in opposite directions)
Exponents (Powers)
- $0^0$ (Base pulls to 0; Exponent pulls to 1)
- $1^\infty$ (Base pulls to 1; Exponent pulls to $\infty$ or 0)
- $\infty^0$ (Base pulls to $\infty$; Exponent pulls to 1)

Different function pairs can produce different limits for the same symbolic form.

Correction note: in standard limit notation, $0/0$ means numerator and denominator both tend to $0$, and $\infty/\infty$ means both tend to $\infty$.

Indeterminate Form Example: $1^\infty$

Let \[ f_n = 1+\frac{1}{n^x}, \qquad g_n=n^y \] with positive integers $x,y$. Then \[ \lim_{n\to\infty}\left(1+\frac{1}{n^x}\right)^{n^y} \] has type $1^\infty$.

Use logs: \[ \log L_n=n^y\log(1+n^{-x})\sim n^y\cdot n^{-x}=n^{y-x}. \]

So the result depends on $x,y$:

$x=y \Rightarrow L_n\to e$
$x>y \Rightarrow L_n\to 1$
$x<y \Rightarrow L_n\to\infty$

$0/0$ and L’Hospital’s Rule

If $f(x)\to 0$ and $g(x)\to 0$ as $x\to a$, the quotient is indeterminate.
L’Hospital’s rule compares rates of change: \[ \lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}, \] when the rule’s conditions are satisfied.

This turns a static “zero divided by zero” into a dynamic comparison of speeds.

Example: \[ f(x)=\frac{1}{x},\quad g(x)=\frac{1}{x^2},\quad x\to\infty. \] Then \[ f'(x)=-\frac{1}{x^2},\quad g'(x)=-\frac{2}{x^3}, \] and \[ \frac{f'(x)}{g'(x)}=\frac{x}{2}\to\infty. \] Hence \[ \lim_{x\to\infty}\frac{(1/x)}{(1/x^2)}=\infty. \]

Continuous Function

A function $f$ is continuous at $x=a$ if \[ \lim_{x\to a}f(x)=f(a). \]

Formal $\epsilon$-$\delta$ definition:
for every $\epsilon>0$, there exists $\delta>0$ such that \[ |x-a|<\delta \Longrightarrow |f(x)-f(a)|<\epsilon. \]

A Famous Non-Continuous Function

\[ f(x)=\sin\left(\frac{1}{x}\right) \] is not continuous at $x=0$ because it oscillates infinitely between $-1$ and $1$ as $x\to 0$, so the limit does not exist.

Connection Between Limit and Continuity

Continuity means: as we shrink the neighborhood around $a$, the limit agrees with the actual value at $a$.
When this local condition holds at every point in a domain, the function is continuous on that domain.

Takeaway. Limits describe approach, and continuity requires that approach value to match function value.

--- title: "Gilbert Strang's Calculus: Limits and Continuous Functions" author: "Chao Ma" date: "2026-02-28" categories: [Calculus, Limits, Continuity, Indeterminate Forms, L'Hospital Rule, Gilbert Strang] --- {{< video https://www.youtube.com/watch?v=kAv5pahIevE&list=PLFW_V3qDH5jRyfpD9uiq6aKVTWIxpbIm3&index=9 >}} # Limit Definition Limits are intuitive ("approach a value"), but calculus needs a strict definition. For a sequence $a_n$, write $$ \lim_{n\to\infty} a_n = A $$ if for every $\epsilon>0$, there exists an integer $N$ such that for all $n>N$, $$ |a_n-A|<\epsilon. $$ This is the $\epsilon$-$N$ definition: eventually, all terms stay within any target tolerance around $A$. ![Epsilon neighborhood for sequence limits](limits-continuous-functions-1.png) # Properties of Limit If $\lim_{n\to\infty} a_n=A$ and $\lim_{n\to\infty} b_n=B$, then: - $a_n \pm b_n \to A \pm B$ - $a_n b_n \to AB$ - $a_n / b_n \to A/B$ when $B\neq 0$ - $(a_n)^{b_n} \to A^B$ in regular cases ## Dangerous! For these properties, special endpoint cases become indeterminate forms: - In subtraction, if both limits are $\infty$, then $\infty-\infty$ is indeterminate. - In multiplication, if limits are $0$ and $\infty$, then $0\cdot\infty$ is indeterminate. - In division: - if both limits are $0$, then $0/0$ is indeterminate. - if both limits are $\infty$, then $\infty/\infty$ is indeterminate. - In powers: - if base and exponent limits are both $0$, then $0^0$ is indeterminate. - if base tends to $1$ and exponent tends to $\infty$, then $1^\infty$ is indeterminate. - if base tends to $\infty$ and exponent tends to $0$, then $\infty^0$ is indeterminate. Here are the 7 indeterminate forms (grouped as in your note): - **Quotients (Fractions)** - $0/0$ (Numerator pulls to 0; Denominator pulls to $\infty$) - $\infty/\infty$ (Numerator pulls to $\infty$; Denominator pulls to 0) - **Products (Multiplication)** - $0\cdot\infty$ (One factor pulls to 0; the other pulls to $\infty$) - **Differences (Subtraction)** - $\infty-\infty$ (Both terms pull the result in opposite directions) - **Exponents (Powers)** - $0^0$ (Base pulls to 0; Exponent pulls to 1) - $1^\infty$ (Base pulls to 1; Exponent pulls to $\infty$ or 0) - $\infty^0$ (Base pulls to $\infty$; Exponent pulls to 1) Different function pairs can produce different limits for the same symbolic form. Correction note: in standard limit notation, $0/0$ means numerator and denominator both tend to $0$, and $\infty/\infty$ means both tend to $\infty$. ![Seven indeterminate forms](limits-continuous-functions-2.png) ## Indeterminate Form Example: $1^\infty$ Let $$ f_n = 1+\frac{1}{n^x}, \qquad g_n=n^y $$ with positive integers $x,y$. Then $$ \lim_{n\to\infty}\left(1+\frac{1}{n^x}\right)^{n^y} $$ has type $1^\infty$. Use logs: $$ \log L_n=n^y\log(1+n^{-x})\sim n^y\cdot n^{-x}=n^{y-x}. $$ So the result depends on $x,y$: - $x=y \Rightarrow L_n\to e$ - $x>y \Rightarrow L_n\to 1$ - $x<y \Rightarrow L_n\to\infty$ ![Case split for $(1+1/n^x)^{n^y}$](limits-continuous-functions-3.png) # $0/0$ and L'Hospital's Rule If $f(x)\to 0$ and $g(x)\to 0$ as $x\to a$, the quotient is indeterminate. L'Hospital's rule compares rates of change: $$ \lim_{x\to a}\frac{f(x)}{g(x)} = \lim_{x\to a}\frac{f'(x)}{g'(x)}, $$ when the rule's conditions are satisfied. This turns a static "zero divided by zero" into a dynamic comparison of speeds. Example: $$ f(x)=\frac{1}{x},\quad g(x)=\frac{1}{x^2},\quad x\to\infty. $$ Then $$ f'(x)=-\frac{1}{x^2},\quad g'(x)=-\frac{2}{x^3}, $$ and $$ \frac{f'(x)}{g'(x)}=\frac{x}{2}\to\infty. $$ Hence $$ \lim_{x\to\infty}\frac{(1/x)}{(1/x^2)}=\infty. $$ # Continuous Function A function $f$ is continuous at $x=a$ if $$ \lim_{x\to a}f(x)=f(a). $$ ![Continuity as "no jump" at a point](limits-continuous-functions-4.png) Formal $\epsilon$-$\delta$ definition: for every $\epsilon>0$, there exists $\delta>0$ such that $$ |x-a|<\delta \Longrightarrow |f(x)-f(a)|<\epsilon. $$ ![Epsilon-delta neighborhood view](limits-continuous-functions-5.png) ## A Famous Non-Continuous Function $$ f(x)=\sin\left(\frac{1}{x}\right) $$ is not continuous at $x=0$ because it oscillates infinitely between $-1$ and $1$ as $x\to 0$, so the limit does not exist. # Connection Between Limit and Continuity Continuity means: as we shrink the neighborhood around $a$, the limit agrees with the actual value at $a$. When this local condition holds at every point in a domain, the function is continuous on that domain. ![Limit and continuity connection map](limits-continuous-functions-6.png) --- **Takeaway.** Limits describe approach, and continuity requires that approach value to match function value.