Limit

Finite Limit

Definitions

• The limit of $f$, as $x$ approaches $x_0$, is $L$
• $\underset{x\to x_0}{\lim }f(x)=L$
• ((ε, δ)-definition of limit. Cauchy)
  • $\forall \epsilon >0,\exists \delta >0,\forall x\in \mathbb{R},(0<|x-x_0|<\delta ⟹|f(x)-L|<\epsilon )$
• (Sequential Criterion for a Limit of a Function. Heine)
  • $\forall (x_n{)}_{n=1}^{\infty },((\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n\ne x_0))⟹\underset{n\to \infty }{\lim }f(x_n)=L$
• (left & right-hand limit) (4.48)
  • $\underset{x\to {x_0}^{-}}{\lim }f(x)=\underset{x\to {x_0}^{+}}{\lim }f(x)=L$
• (4.40) $\underset{h\to 0}{\lim }f(x_0+h)=L$

Theorems

• (4.31) Uniqueness of the limit of function

  • $(f(x)\underset{n\to x_0}{⟶}L\wedge f(x)\underset{n\to x_0}{⟶}M)⟹L=M$

• (4.33) Limit of Linear Function

  • Let $f$ a linear function then for each $x_0\in \mathbb{R}$, we have $\underset{x\to x_0}{\lim }f(x)=f(x_0)$

• $\forall \epsilon >0,\exists \delta >0:\forall x\in \mathbb{R},(0<|x-x_0|<\delta ⟹|f(x)-L|<K\epsilon )⟹\underset{x\to x_0}{\lim }f(x)=L$ (given $K$ is a constant positive real number)

• given $\underset{x\to x_0}{\lim }f(x)\in \mathbb{R}$

  • (4.34, Local Property) $\exists \delta :\forall x\in (-\delta ,x_0)\cup (x_0,\delta ),f(x)=g(x)⟹\underset{x\to x_0}{\lim }f(x)=\underset{x\to x_0}{\lim }g(x)$

• $f$ is defined on $(-{\delta }_0,x_0)\cup (x_0,{\delta }_0)$

  • (4.36) $0\ne \underset{x\to x_0}{\lim }f(x)\in \mathbb{R}⟹\exists \delta :\forall x\in (-\delta ,x_0)\cup (x_0,\delta ),f(x)\ne 0$

• (4.43) Squeeze theorem for functions

  • $(f(x)\le g(x)\le h(x))\wedge (\underset{x\to x_0}{\lim }f(x)=\underset{x\to x_0}{\lim }h(x)=L)⟹\underset{x\to x_0}{\lim }g(x)=L$

• given the limits of $f$ and $g$ is defined (finite or infinite)

  • (4.41) $\exists \epsilon >0:\forall x\in N_{\epsilon }^{\ast }(x_0),f(x)\le g(x)⟹\underset{x\to x_0}{\lim }f(x)\le \underset{x\to x_0}{\lim }g(x)$
  • (4.42) $\exists \epsilon >0:\forall x\in N_{\epsilon }^{\ast }(x_0),f(x)<g(x)⟸\underset{x\to x_0}{\lim }f(x)<\underset{x\to x_0}{\lim }g(x)$

Arithmetic

Assuming $f,g$ are defined on $N_{\epsilon }^{\ast }(x_0)$, and their limits are exist

Limit Laws
Sum / Difference	$\underset{x\to x_0}{\lim }(f(x)\pm g(x))=\underset{x\to x_0}{\lim }f(x)\pm \underset{x\to x_0}{\lim }g(x)$
Product	$\underset{x\to x_0}{\lim }(f(x)\cdot g(x))=\underset{x\to x_0}{\lim }f(x)\cdot \underset{x\to x_0}{\lim }g(x)$
	$\underset{x\to x_0}{\lim }(f(x)\cdot c)=\underset{x\to x_0}{\lim }f(x)\cdot c$
	$\underset{x\to x_0}{\lim }(f(x){)}^n=(\underset{x\to x_0}{\lim }f(x){)}^n$
Quotient	$\underset{x\to x_0}{\lim }(f/g)(x)=\underset{x\to x_0}{\lim }f(x)/\underset{x\to x_0}{\lim }g(x)$ where $\underset{x\to x_0}{\lim }g(x)\ne 0$
Composite Function (4.39)	$\underset{x\to x_0}{\lim }f(x)=\underset{t\to t_0}{\lim }f(g(t))$	$\underset{t\to t_0}{\lim }g(t)=x_0$ is defined and ($f$ is continuous at $x_0$ or $g(t)\ne x_0$ for some deleted neighborhood of $t_0$)

• (q4.62) $\underset{x\to 0}{\lim }f(x)=L⟹\underset{x\to 0}{\lim }f(x^2)=L$
• (q4.63) $\underset{x\to 0}{\lim }f(x)=\underset{x\to 0}{\lim }f(xk)$ (where $k\ne 0$ and the limits exists)
• $\underset{x\to \infty }{\lim }f(x)=0\wedge \exists M:|g(x)|<M⟹\underset{x\to \infty }{\lim }f(x)g(x)=0$ (analogously 2.22)
• $\underset{x\to x_0}{\lim }f(x)=0⟺\underset{x\to x_0}{\lim }|f(x)|=0$ (analogously q2.20a)

for $\underset{x\to x_0}{\lim }f(x{)}^{g(x)}$ you can use in these methods:

• Heine’s Sequential Criterion with t6.15
• Using in the identity $f(x{)}^{g(x)}=e^{g(x)\ln (f(x))}$ (where $0<f(x{)}^{g(x)}$) and using in Composite Function (5.14)

One-Sided Limits

• $L$ is the right-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{+}}{\lim }f(x)=L$
  • $(\forall \epsilon >0)(\exists \delta >0)(\forall x\in \mathbb{R})(x\in (x_0,x_0+\delta )⟹f(x)\in (L-\epsilon ,L+\epsilon ))$
  • Sequential Criterion (Heine)
    • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n>x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=L$
• $L$ is the left-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{-}}{\lim }f(x)=L$
  • $(\forall \epsilon >0)(\exists \delta >0)(\forall x\in \mathbb{R})(x\in (x_0-\delta ,x_0)⟹f(x)\in (L-\epsilon ,L+\epsilon ))$
  • Sequential Criterion (Heine)
    • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n<x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=L$

Infinite limits

• $\underset{x\to x_0}{\lim }f(x)=\infty$
• $(\forall M\in \mathbb{R})(\exists \delta >0)(\forall x\in \mathbb{R})(0<|x-x_0|<\delta ⟹f(x)>M)$
  • (Note: It’s sufficient to ensure for $M>0$.)
• $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n\ne x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=\infty$
• $\underset{x\to {x_0}^{-}}{\lim }f(x)=\infty =\underset{x\to {x_0}^{+}}{\lim }f(x)$

$\underset{x\to x_0}{\lim }f(x)=-\infty$ (or one-sided Limits) defined the same, just replace $f(x)>m$ with $f(x)<m$

One-Sided Limits

• $\infty$ is the right-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{+}}{\lim }f(x)=\infty$
  • $(\forall M\in \mathbb{R})(\exists \delta >0)(\forall x\in \mathbb{R})(x\in (x_0,x_0+\delta )⟹f(x)>M)$
  • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n>x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=\infty$
• $\infty$ is the left-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{-}}{\lim }f(x)=\infty$
  • $(\forall M\in \mathbb{R})(\exists \delta >0)(\forall x\in \mathbb{R})(x\in (x_0-\delta ,x_0)⟹f(x)>M)$
  • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n<x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=\infty$
• $-\infty$ is the right-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{+}}{\lim }f(x)=-\infty$
  • $(\forall M\in \mathbb{R})(\exists \delta >0)(\forall x\in \mathbb{R})(x_0<x<x_0+\delta )⟹f(x)<M)$
  • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n>x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=-\infty$
• $-\infty$ is the left-hand limit of $f$ at $x_0$
  • $\underset{x\to {x_0}^{-}}{\lim }f(x)=-\infty$
  • $(\forall M\in \mathbb{R})(\exists \delta >0)(\forall x\in \mathbb{R})(x_0-\delta <x<x_0)⟹f(x)<M)$
  • $\forall (x_n{)}_{n=1}^{\infty }(\underset{n\to \infty }{\lim }(x_n)=x_0)\wedge (\forall n\in \mathbb{N},x_n<x_0)⟹\underset{n\to \infty }{\lim }f(x_n)=-\infty$

Arithmetic of Infinite limits

• $\infty +\infty =\infty$
• $\infty +r=\infty$
• $\infty \cdot \infty =\infty$
• $\infty \cdot r_{>0}=\infty$
• $1/\infty =0$
• $1/0^{+}=\infty$
• $\underset{x\to x_0}{\lim }(f(x))=\infty ⟺\underset{x\to x_0}{\lim }(-f(x))=-\infty$
• Squeeze Theorem for infinite limit
  • $f(x)\to \infty \wedge \exists M:\forall x>M,g(x)\ge f(x)⟹g(x)\to \infty$ (analogously to 2.45)

Limits at Infinity

• $L$ is the limit of $f$

  • at $\infty$
    • $\underset{x\to \infty }{\lim }f(x)=L$
    • $\forall \epsilon >0,\exists M:\forall x\in \mathbb{R},(x>M⟹|f(x)-L|<\epsilon )$
    • $\forall (x_n{)}_{n=1}^{\infty }\underset{n\to \infty }{\lim }{x}_n=\infty ⟹\underset{n\to \infty }{\lim }f(x_n)=L$
  • at $-\infty$
    • $\underset{x\to -\infty }{\lim }f(x)=L$
    • $(\forall \epsilon >0)(\exists M)(\forall x\in \mathbb{R})(x<M⟹|f(x)-L|<\epsilon )$
    • $\forall (x_n{)}_{n=1}^{\infty }\underset{n\to \infty }{\lim }{x}_n=-\infty ⟹\underset{n\to \infty }{\lim }f(x_n)=L$

• $\infty$ is the limit of $f$

  • at $\infty$
    • $\underset{x\to \infty }{\lim }f(x)=\infty$
    • $(\forall M_1\in \mathbb{R})(\exists M_2\in \mathbb{R})(\forall x\in \mathbb{R})(x>M_2⟹f(x)>M_1)$
    • $\forall (x_n{)}_{n=1}^{\infty }\underset{n\to \infty }{\lim }{x}_n=\infty ⟹\underset{n\to \infty }{\lim }f(x_n)=\infty$
  • at $-\infty$todo

• $-\infty$ is the limit of $f$

  • at $\infty$todo
  • at $-\infty$todo

---

• Limit at Infinity of Rational Function $\frac{f(x)}{g(x)}=\frac{a_n{x}^n+\cdots +a_0}{b_m{x}^m+\cdots +b_0}$
  • $\mathrm{deg}(f)<\mathrm{deg}(g)⟹\underset{x\to \infty }{\lim }\frac{\frac{a_n{x}^n}{x^m}+...+\frac{a_0}{x^m}}{\frac{b_m{x}^m}{x^m}+...+\frac{b_0}{x^m}}=\frac{0+...+0}{b_m+...+0}=\frac{0}{b_m}=0$
  • $\mathrm{deg}(f)=\mathrm{deg}(g)⟹\underset{x\to \infty }{\lim }\frac{\frac{a_n{x}^n}{x^n}+...+\frac{a_0}{x^n}}{\frac{b_n{x}^n}{x^n}+...+\frac{b_0}{x^n}}=\frac{a_n+...+0}{b_n+...+0}=\frac{a_n}{b_n}$
  • $\mathrm{deg}(f)>\mathrm{deg}(g)⟹\underset{x\to \infty }{\lim }\frac{f(x)}{g(x)}=\pm \infty$ (depending on the sign of $\frac{a_n}{b_m}$)

L’Hôpital’s rule

• $x_0\in I$ is finite, $\infty$ or $-\infty$
• $I$ is an open interval with $x_0$ (for two-sided limits) or an open interval with endpoint $x_0$ (for one-sided limits or limits at infinity if $x_0$ is infinite).
• $f$ and $g$ are differentiable on $I$ (except possibly at $x_0$),
• $g^{\prime }(x)\ne 0$ on $I$ (except possibly at $x_0$)
• $L=\underset{x\to x_0}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$ exists (finite, $\infty$ or $-\infty$)
• $\underset{x\to x_0}{\lim }f(x)=\underset{x\to x_0}{\lim }g(x)=0$ or $\underset{x\to x_0}{\lim }|f(x)|=\underset{x\to x_0}{\lim }|g(x)|=\infty$ $\underset{x\to x_0}{\lim }\frac{f(x)}{g(x)}=\underset{x\to x_0}{\lim }\frac{f^{\prime }(x)}{g^{\prime }(x)}$

Indeterminate form

$\frac{0}{0},\frac{\infty }{\infty },0\times \infty ,\infty -\infty ,0^0,1^{\infty },{\infty }^0$

		Transformation to $\frac{0}{0}$	Transformation to $\frac{\infty }{\infty }$
$\underset{x\to x_0}{\lim }\frac{f(x)}{g(x)}$	$\frac{0}{0}$	-	$\underset{x\to x_0}{\lim }\frac{1/g(x)}{1/f(x)}$
	$\frac{\infty }{\infty }$	$\underset{x\to x_0}{\lim }\frac{1/g(x)}{1/f(x)}$	-
$\underset{x\to x_0}{\lim }f(x)-g(x)$	$\infty -\infty$		$\ln \underset{x\to x_0}{\lim }\frac{e^{f(x)}}{e^{g(x)}}$
$\underset{x\to x_0}{\lim }f(x)\cdot g(x)$	$0\cdot \infty$
$\underset{x\to x_0}{\lim }f(x{)}^{g(x)}$	$1^{\infty }$	$\exp \underset{x\to x_0}{\lim }\frac{\ln f(x)}{1/g(x)}$	$\exp \underset{x\to x_0}{\lim }\frac{g(x)}{1/\ln f(x)}$
	$0^0$
	${\infty }^0$

Bounded Monotonic Function

• (5.39) Bounded Monotonic Function - $f$ is monotonic and defined on open interval $(a,b)$.
  • $f$ is bounded on an open interval $(a,b)$, then both $\underset{x\to b^{-}}{\lim }f(x)$ and $\underset{x\to a^{+}}{\lim }f(x)$ exist (finite)
    • if $f$ is increasing then
      • $\underset{x\to b^{-}}{\lim }f(x)=\sup f((a,b))$
      • $\underset{x\to a^{+}}{\lim }f(x)=\inf f((a,b))$
    • if $f$ is decreasing then
      • $\underset{x\to b^{-}}{\lim }f(x)=\inf f((a,b))$
      • $\underset{x\to a^{+}}{\lim }f(x)=\sup f((a,b))$
  • if $f$ is not bounded above
    • if $f$ is increasing then
      • $\underset{x\to b^{-}}{\lim }f(x)=+\infty$
    • if $f$ is decreasing then
      • $\underset{x\to a^{+}}{\lim }f(x)=+\infty$
  • if $f$ is not bounded below
    • if $f$ is increasing then
      • $\underset{x\to a+}{\lim }f(x)=-\infty$
    • if $f$ is decreasing then
      • $\underset{x\to b^{-}}{\lim }f(x)=-\infty$

Examples

Trigonometric functions

• (4.44a) $\underset{x\to 0}{\lim }\sin x=0$

• (4.44b) $\underset{x\to 0}{\lim }\cos x=1$

• (4.45) $\underset{x\to 0}{\lim }\frac{\sin x}{x}=\underset{x\to 0}{\lim }\frac{x}{\sin x}=1$

• $\underset{x\to 0}{\lim }\frac{\tan x}{x}=1$

• $\underset{x\to 0}{\lim }\frac{{\tan }^{2}x}{x}=0$

• $\underset{x\to 0}{\lim }\frac{\sin (cx)}{x}=c$

• $\underset{x\to 0}{\lim }\frac{\sin (x)}{cx}=\frac{1}{c}$

• $\underset{x\to 0}{\lim }\frac{\arcsin x}{x}=\underset{x\to 0}{\lim }\frac{\arctan x}{x}=1$

• $\underset{x\to \infty }{\lim }(x\sin (1/x))=1$

• $(\cos x{)}^{\sin x}$

  • $\underset{x\to 0}{\lim }(\cos x{)}^{\sin x}=$todo see e6.3 page 179

Exponential functions

Functions of the form c^g(x)

$\lim c^x$

(6.9)

$c^x$	$x\to -\infty$	$x\to x_0$	$x\to \infty$
$0<c<1$	$\lim =\infty$		$\lim =0$
$c=1$	$\lim =1$	$\lim =1$	$\lim =1$
$1<c$	$\lim =0$	$\lim =c^{x_0}$	$\lim =\infty$

$\lim c^{1/x}$

$\underset{x\to \infty }{\lim }\sqrt[x]{c}=\underset{x\to \infty }{\lim }{c}^{1/x}=\{\begin{array}{ll}1,& c>0\\ 0,& c=0\\ \text{does not exist},& c<0\end{array}$

Functions of the form $f(x{)}^{g(x)}$

• (6.16) $\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e$

• (6.17) $\underset{x\to -\infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e$

• (6.18) $\underset{x\to 0}{\lim }(1+x{)}^{1/x}=e$

• $\underset{x\to \infty }{\lim }{\left(1-\frac{1}{x}\right)}^x=\frac{1}{e}$

Polynomials

• $\underset{x\to c}{\lim }p(x)=p(c)$
• $p$ is n degree polynomial and $a_n\ne 0$
  • $\underset{x\to \infty }{\lim }p(x)=\text{sign}(a_n)\cdot \infty$
  • $\underset{x\to -\infty }{\lim }p(x)=\{\begin{array}{ll}\mathrm{sign}(a_n)\infty & n\text{ even}\\ -\mathrm{sign}(a_n)\infty & n\text{ odd}\end{array}$

Functions of the form x^c

• (e4.31) $\underset{x\to \infty }{\lim }\sqrt{x}=\underset{x\to \infty }{\lim }{x}^{1/2}=\infty$

• (q4.82a) $\underset{x\to \infty }{\lim }x=\infty$

• $\underset{x\to x_0}{\lim }x=x_0$

• $\underset{x\to x_0}{\lim }{x}^c=x_0^c$

• $c>0$

  • $\underset{x\to \infty }{\lim }{x}^c=\infty$
  • $\underset{x\to 0^{+}}{\lim }{x}^c=0$

Logarithmic functions

b<1

• $\underset{x\to \infty }{\lim }{\log }_{b}x=-\infty$
• $\underset{x\to 0^{+}}{\lim }{\log }_{b}x=\infty$

b>1

• $\underset{x\to \infty }{\lim }{\log }_{b}x=\infty$
• $\underset{x\to 0^{+}}{\lim }{\log }_{b}x=-\infty$

Natural logarithms

• $\underset{x\to x_0}{\lim }\ln x=\ln x_0$
• $\underset{x\to \infty }{\lim }\ln x=\infty$
• $\underset{x\to 0^{+}}{\lim }\ln x=-\infty$

Other

• (q4.85c) $\underset{x\to \infty }{\lim }f(x)=\underset{x\to 0^{+}}{\lim }f(1/x)$ (assuming $f$ defined on $(M,\infty )$)

• (q4.85d) $\underset{x\to -\infty }{\lim }f(x)=\underset{x\to \infty }{\lim }f(-x)$ (assuming $f$ defined on $(-\infty ,M)$)

• $\underset{x\to 0^{+}}{\lim }{x}^x=1$

• $\frac{1}{x-x_0}$

  • $\underset{x\to x_0}{\lim }\frac{1}{x-x_0}$ does not exist $⟸\underset{x\to x_0^{+}}{\lim }\frac{1}{x-x_0}=\infty \wedge \underset{x\to x_0^{-}}{\lim }\frac{1}{x-x_0}=-\infty$
  • $\underset{x\to \infty }{\lim }\frac{1}{x-x_0}=0$
  • $\frac{1}{x}$ (Special Case $x_0=0$)
    • $\underset{x\to 0}{\lim }\frac{1}{x}$ does not exist $⟸\underset{x\to 0^{+}}{\lim }\frac{1}{x}=\infty \wedge \underset{x\to 0^{-}}{\lim }\frac{1}{x}=-\infty$
    • (q4.78) $\underset{x\to \infty }{\lim }\frac{1}{x}=0$

• $\frac{x^r}{a^x}$ (assuming $a>1$, and $r\in \mathbb{R}$)

  • (q6.13) $\underset{x\to \infty }{\lim }\frac{x^r}{a^x}=0$

• (q6.17) $\underset{x\to \infty }{\lim }\frac{\ln x}{x^a}=0$ where $a>0$

Asymptotes

• The vertical line $x=x_0$ is a vertical asymptote of the function $y=f(x)$ if

  • $\underset{x\to x_0^{-}}{\lim }f(x)=\pm \infty$, or
  • $\underset{x\to x_0^{+}}{\lim }f(x)=\pm \infty$

• The horizontal line $y=c$ is a horizontal asymptote of the function $y=f(x)$ if

  • $\underset{x\to -\infty }{\lim }f(x)=c$, or
  • $\underset{x\to +\infty }{\lim }f(x)=c$

• The straight line $y=mx+n$ is an oblique asymptote of the function $y=f(x)$ if

  • $\underset{x\to -\infty }{\lim }\left[f(x)-(mx+n)\right]=0$, or
  • $\underset{x\to +\infty }{\lim }\left[f(x)-(mx+n)\right]=0$