Limits of Real Functions

Real One-Sided Limits

Definition: Left-Sided Limit

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

A number $L\in \mathbb{R}$ is called the left-sided limit of $f$ as $x$ approaches $c\in D$ if for every $\epsilon >0$ there is some $\delta >0$ such that for all $x\in D$ with $x<c$

$$|x-c|<\delta ⟹|f(x)-L|<\epsilon$$

NOTATION

$$\begin{array}{rl}& \underset{x\to c^{-}}{\lim }f(x)=L\\ & \underset{\begin{array}{c}x\to c^{+}\\ x<c\end{array}}{\lim }f(x)=L\end{array}$$

Definition: Right-Sided Limit

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

A number $L\in \mathbb{R}$ is called the right-sided limit of $f$ as $x$ approaches $c\in D$ if for every $\epsilon >0$ there is a $\delta >0$ such that for all $x\in D$ with $x>c$

$|x-c|<\delta ⟹|f(x)-L|<\epsilon$

NOTATION

$$\begin{array}{rl}& \underset{x\to c^{+}}{\lim }f(x)=L\\ & \underset{\begin{array}{c}x\to c^{+}\\ x>c\end{array}}{\lim }f(x)=L\end{array}$$

Real Limits

Definition: Limit of a Function (Cauchy)

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

A real number $L\in \mathbb{R}$ is called the limit of $f$ as $x$ approaches $c\in \mathbb{R}$ if for every $\epsilon >0$ there is a $\delta >0$ such that for all $x\in D$ with $x\ne c$

$$|x-c|<\delta ⟹|f(x)-L|<\epsilon$$

NOTATION

$$\underset{x\to c}{\lim }f(x)=L$$

Definition: Limit at Positive Infinity

A real number $L\in \mathbb{R}$ is called the limit of $f$ as $x$ approaches positive infinity if for every $\epsilon >0$ there is a $A\in \mathbb{R}$ such that

$$|f(x)-L|<\epsilon \forall x\ge A$$

NOTATION

$$\underset{x\to \infty }{\lim }f(x)=L$$

Definition: Limit at Negative Infinity

A real number $L\in \mathbb{R}$ is called the limit of $f$ as $x$ approaches negative infinity if for every $\epsilon >0$ there is a $A\in \mathbb{R}$ such that

$$|f(x)-L|<\epsilon \forall x\le A$$

NOTATION

$$\underset{x\to -\infty }{\lim }f(x)=L$$

If $f$ has a limit $L\in \mathbb{R}$, then the limit is said to exist.

Theorem: Real Limit and One-Sided Real Limits

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The two-sided limit of $f$ as $x$ approaches $c\in \mathbb{R}$ exists if and only if both two-sided limit of $f$ at $c$ exist and are equal.

$$\underset{x\to c}{\lim }f(x)=L\in \mathbb{R}⟺\underset{x\to c^{-}}{\lim }f(x)=\underset{x\to c^{+}}{\lim }f(x)=L$$

PROOF

TODO

Theorem: Limits through Transformations

Let $f$ and $g$ be real functions and let $c\in \mathbb{R}$.

If there exists some deleted neighborhood of $c$ on which $f$ and $g$ are equal and the limit of $g$ at $c$ is $L\in \mathbb{R}$, then

$$\underset{x\to c}{\lim }f(x)=\underset{x\to c}{\lim }g(x)=L.$$

NOTE

This theorem is extremely powerful because it allows us to find the limit of $f$ by finding another function $g$ (usually through algebraic manipulations) whose limit is easier to compute, so long as the two functions are equal around $c$. We don’t care about what happens at $x=c$ or if $f$ and $g$ are even defined at $c$.

PROOF

Let $\epsilon >0$ be any positive real number.

Since there is some deleted neighborhood of $c$ on which $f$ and $g$ are equal, we know that there exists some ${\delta }_1>0$ such that

$$0<|x-c|<{\delta }_1⟹f(x)=g(x).$$

Since ${\lim }_{x\to c}g(x)=L$, we know that for our choice of $\epsilon$ there exists some ${\delta }_2>0$ such that

$$0<|x-c|<{\delta }_2⟹|g(x)-L|<\epsilon .$$

Set $\delta =\min \{{\delta }_1,{\delta }_2\}$.

Since $\delta \le {\delta }_1$, we have

$$0<|x-c|<\delta ⟹0<|x-c|<\delta$$

and, since $0<|x-c|<{\delta }_1⟹f(x)=g(x)$, we have proved that

$$0<|x-c|<\delta ⟹f(x)=g(x).$$

Similarly, since $\delta \le {\delta }_2$, we have

$$0<|x-c|<\delta ⟹0<|x-c|<\delta$$

and, since $0<|x-c|<{\delta }_2⟹|g(x)-L|<\epsilon$, we have proved that

$$0<|x-c|<\delta ⟹|g(x)-L|<\epsilon .$$

Now we combine these facts. Consider $|f(x)-L|$. For all $x$ with $0<|x-c|<\delta$, we showed that $f(x)=g(x)$. Therefore,

$$0<|x-c|<\delta ⟹|f(x)-L|=|g(x)-L|$$

Moreover, we showed that $|g(x)-L|<\epsilon$ for all $x$ with $0<|x-c|<\delta$. Therefore,

$$0<|x-c|<\delta ⟹|f(x)-L|<\epsilon .$$

Theorem: Arithmetic with Real Limits

Let $f$ and $g$ be real functions and let $c\in \mathbb{R}\cup \{-\infty ,+\infty \}$.

If the limits of $f$ and $g$ exist at $c$, then

$$\begin{array}{rl}& \underset{x\to c}{\lim }\left(\alpha f(x)+\beta g(x)\right)=\alpha \underset{x\to c}{\lim }f(x)+\beta \underset{x\to c}{\lim }g(x)\forall \alpha ,\beta \in \mathbb{R}\\ & \\ & \underset{x\to c}{\lim }\left(f(x)g(x)\right)=\left(\underset{x\to c}{\lim }f(x)\right)\cdot \left(\underset{x\to c}{\lim }g(x)\right)\\ & \\ & \underset{x\to c}{\lim }\frac{f(x)}{g(x)}=\frac{\underset{x\to c}{\lim }f(x)}{\underset{x\to c}{\lim }g(x)},\text{ provided that }\underset{x\to c}{\lim }g(x)\ne 0\\ & \underset{x\to c}{\lim }f(x{)}^{g(x)}=\underset{x\to c}{\lim }f(x{)}^{\underset{x\to c}{\lim }g(x)}\end{array}$$

PROOF

TODO

WARNING

These do not apply to infinite limits.

Infinite Limits

Definition: Infinite Limits

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

Definition: Approaching Positive Infinity

We say that $f$ approaches positive infinity as $x$ approaches $c\in \mathbb{R}$ if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$

$$|x-c|<\delta ⟹f(x)>A$$

NOTATION

$$\underset{x\to c}{\lim }f(x)=\infty$$

Definition: Approaching Positive Infinity for $x\to \pm \infty$

We say that $f$ approaches positive infinity

• as $x$ approaches $-\infty$ if for every $A\in \mathbb{R}$ there is a $B\in \mathbb{R}$ such that $f(x)>A$ for all $x<B$;

• as $x$ approaches $\infty$ if for every $A\in \mathbb{R}$ there is a $B\in \mathbb{R}$ such that $f(x)>A$ for all $x>B$.

NOTATION

$$\underset{x\to -\infty }{\lim }f(x)=\infty$$ $$\underset{x\to \infty }{\lim }f(x)=\infty$$

Definition: Approaching Negative Infinity

We say that $f$ approaches negative infinity as $x$ approaches $c\in \mathbb{R}$ if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$

$$|x-c|<\delta ⟹f(x)<A$$

NOTATION

$$\underset{x\to c}{\lim }f(x)=-\infty$$

Definition: Approaching Negative Infinity for $x\to \pm \infty$

We say that $f$ approaches negative infinity

• as $x$ approaches $-\infty$ if for every $A\in \mathbb{R}$ there is a $B\in \mathbb{R}$ such that $f(x)<A$ for all $x<B$;

• as $x$ approaches $\infty$ if for every $A\in \mathbb{R}$ there is a $B\in \mathbb{R}$ such that $f(x)<A$ for all $x>B$.

NOTATION

$$\underset{x\to -\infty }{\lim }f(x)=-\infty$$ $$\underset{x\to \infty }{\lim }f(x)=-\infty$$

Infinite One-Sided Limits

Definition: Infinite One-Sided Limits

Let $f$ be a real function.

Definition: Positive Infinity as a Left-Sided Limit

We say that $f$ approaches positive infinity as $x$ approaches $c\in D$ from the left if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$ with $x<c$

$$|x-c|<\delta ⟹f(x)>A$$

NOTATION

$$\underset{x\to c^{-}}{\lim }f(x)=\infty$$

Definition: Negative Infinity as a Left-Sided Limit

We say that $f$ approaches negative infinity as $x$ approaches $c\in D$ from the left if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$ with $x<c$

$$|x-c|<\delta ⟹f(x)<A$$

NOTATION

$$\underset{x\to c^{-}}{\lim }f(x)=-\infty$$

Definition: Positive Infinity as a Right-Sided Limit

We say that $f$ approaches positive infinity as $x$ approaches $c\in D$ from the right if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$ with $x>c$

$$|x-c|<\delta ⟹f(x)>A$$

NOTATION

$$\underset{x\to c^{-}}{\lim }f(x)=\infty$$

Definition: Negative Infinity as a Right-Sided Limit

We say that $f$ approaches negative infinity as $x$ approaches $c\in D$ from the right if for every $A\in \mathbb{R}$ there is a $\delta >0$ such that for all $x\in D$ with $x>c$

$$|x-c|<\delta ⟹f(x)<A$$

NOTATION

$$\underset{x\to c^{-}}{\lim }f(x)=-\infty$$

Theorem: Arithmetic with Infinite Limits

Let $f$ and $g$ be real functions and let $c\in \mathbb{R}\cup \{-\infty ,+\infty \}$.

The following rules apply for the limits of $f$ and $g$ at $c$, either real or infinite:

	$\underset{x\to c}{\lim }f(x)=L<0$	$\underset{x\to c}{\lim }f(x)=L>0$
$\underset{x\to c}{\lim }g(x)=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$
$\underset{x\to c}{\lim }g(x)=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$

	$\underset{x\to c}{\lim }f(x)=-\infty$	$\underset{x\to c}{\lim }f(x)=+\infty$
$\underset{x\to c}{\lim }g(x)=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=-\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=?$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$
$\underset{x\to c}{\lim }g(x)=+\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=?$ $\underset{x\to c}{\lim }(f(x)g(x))=-\infty$	$\underset{x\to c}{\lim }(f(x)+g(x))=+\infty$ $\underset{x\to c}{\lim }(f(x)g(x))=+\infty$

NOTE

A question mark (”?”) indicates that we cannot compute the limit directly, but we can try to transform the expression via algebraic manipulations in such a way, so as to make the limit computable.

PROOF

TODO

Theorem: Converting between Limits at Infinity and Limits at Zero

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $L\in \mathbb{R}\cup \{-\infty ,+\infty \}$.

The following rules allow us to convert between limits at infinity and one-sided limits at zero:

$$\underset{x\to +\infty }{\lim }f(x)=L⟺\underset{y\to 0^{+}}{\lim }f\left(\frac{1}{y}\right)=L$$ $$\underset{x\to -\infty }{\lim }f(x)=L⟺\underset{y\to 0^{-}}{\lim }f\left(\frac{1}{y}\right)=L$$

PROOF

TODO

Theorem: Converting between Infinite Limits and Zero Limits

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $c\in \mathbb{R}$.

The following rules allow us to convert between infinite limits and real limits:

• If ${\lim }_{x\to c}f(x)=+\infty$ or ${\lim }_{x\to c}f(x)=-\infty$, then ${\lim }_{x\to c}\frac{1}{f(x)}=0$, i.e.

$$\underset{x\to c}{\lim }f(x)=\pm \infty ⟹\underset{x\to c}{\lim }\frac{1}{f(x)}=0.$$

• If ${\lim }_{x\to c}f(x)=0$ and there exists some neighborhood $\mathcal{N}$ of $c$ on which $f(x)>0$, then ${\lim }_{x\to c}\frac{1}{f(x)}=+\infty$, i.e.

$$\underset{x\to c}{\lim }f(x)=0\text{ and }f(x)>0\text{ for all }x\in \mathcal{N}⟹\underset{x\to c}{\lim }\frac{1}{f(x)}=+\infty .$$

• If ${\lim }_{x\to c}f(x)=0$ and neighborhood $\mathcal{N}$ of $c$ on which $f(x)<0$, then ${\lim }_{x\to c}\frac{1}{f(x)}=-\infty$, i.e.

$$\underset{x\to c}{\lim }f(x)=0\text{ and }f(x)<0\text{ for all }x\in \mathcal{N}⟹\underset{x\to c}{\lim }\frac{1}{f(x)}=-\infty .$$

PROOF

TODO

Theorem: Limits of Compositions

Let $f:{\mathcal{D}}_f\subseteq \mathbb{R}\to \mathbb{R}$ and $g:{\mathcal{D}}_g\subseteq \mathbb{R}\to \mathbb{R}$ be real functions and let $c\in \mathbb{R}$.

If the limit of $g$ exists at $c$ and $f$ is continuous at ${\lim }_{x\to c}g(x)$, then the limit of their composition exists and

$$\underset{x\to c}{\lim }f(g(x))=f\left(\underset{x\to c}{\lim }g(x)\right).$$

PROOF

TODO

Theorem: The Squeeze Theorem for Functions

Let $f:{\mathcal{D}}_f\subseteq \mathbb{R}\to \mathbb{R}$, $g:{\mathcal{D}}_g\subseteq \mathbb{R}\to \mathbb{R}$ and $h:{\mathcal{D}}_h\subseteq \mathbb{R}\to \mathbb{R}$ be real functions and let $c\in \mathbb{R}\cup \{-\infty ,+\infty \}$.

If there exists some neighborhood of $c$ on which $f(x)\le h(x)\le g(x)$ and the limits of $f$ and $g$ exist at $c$ and are equal to the same $L\in \mathbb{R}$, then

$$\underset{c\to x}{\lim }f(x)=\underset{c\to x}{\lim }h(x)=\underset{x\to c}{\lim }g(x)=L.$$

PROOF

TODO

Theorem: Limits Inequality

Let $f:{\mathcal{D}}_f\subseteq \mathbb{R}\to \mathbb{R}$ and $g:{\mathcal{D}}_g\subseteq \mathbb{R}\to \mathbb{R}$ be real functions and let $c\in \mathbb{R}$.

If there exists some neighborhood of $c$ on which $g(x)\le f(x)$ and the limits of $f$ and $g$ exist at $c$, then

$$\underset{x\to c}{\lim }g(x)\le \underset{x\to c}{\lim }f(x)$$

PROOF

TODO

Theorem: L'Hôpital's rule

PROOF

TODO

Theorem

Let $f:{\mathcal{D}}_f\subseteq \mathbb{R}\to \mathbb{R}$ and $g:{\mathcal{D}}_g\subseteq \mathbb{R}\to \mathbb{R}$ be real functions and let $c\in \mathbb{R}$.

If there exists a neighborhood of $c$ on which $f$ is bounded and the limit of $g$ at $c$ is zero, then

$$\underset{x\to c}{\lim }(g(x)f(x))=0.$$

PROOF

TODO

Theorem: Important Trigonometric Limits

Following are some limits involving the real trigonometric functions:

$$\underset{x\to 0}{\lim }\frac{\sin x}{x}=\underset{x\to 0}{\lim }\frac{x}{\sin x}=1$$ $$\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0$$

PROOF

TODO

Theorem: Important Exponential Limits

Following are some limits involving the real exponential function:

$$\underset{x\to -\infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=\underset{x\to +\infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=\mathrm{e}$$ $$\underset{x\to 0}{\lim }(1+x{)}^{\frac{1}{x}}=\mathrm{e}$$

PROOF

TODO

Asymptotes

Definition: Vertical Asymptote

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

We say that $f$ has a vertical asymptote at $x=c\in \mathbb{R}$ if it has at least one infinite one-sided limit, i.e. at least one of the following holds:

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

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

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

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

INTUITION

Intuitively, this definition means that the value of $f(x)$ gets closer and closer to $-\infty$ or $+\infty$ as $x$ approaches $c$ either from the left or from the right.

Definition: Horizontal Asymptote

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

We say that $f$ has a horizontal asymptote $y=a\in \mathbb{R}$ if the limit of $f$ as $x$ approaches positive or negative infinity is $a$, i.e. if at least one of the following holds:

• $\underset{x\to -\infty }{\lim }f(x)=a$

• $\underset{x\to \infty }{\lim }f(x)=a$

INTUITION

Intuitively, this definition means that the value of $f(x)$ gets closer and closer to $a$ as $x$ approaches either positive or negative infinity.

Definition: Oblique Asymptote

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function.

The line $y=ax+b$ is an oblique or slanted asymptote of $f$ if at least one of the following is true:

• $\underset{x\to -\infty }{\lim }[f(x)-(ax+b)]=0$

• $\underset{x\to \infty }{\lim }[f(x)-(ax+b)]=0$

INTUITION

Intuitively, this definition means $f(x)$ gets closer and closer to the line $y=ax+b$ as $x$ approaches either positive or negative infinity.

Theorem: Oblique Asymptotes

Let $f$ be a real function.

The line $y=ax+b$ is an asymptote of $f$ if and only if the limits ${\lim }_{x\to \pm \infty }\frac{f(x)}{x}$ and ${\lim }_{x\to \pm \infty }(f(x)-ax)$ exist and

$$\begin{array}{rl}a& =\underset{x\to \pm \infty }{\lim }\frac{f(x)}{x}\\ & \\ b& =\underset{x\to \pm \infty }{\lim }(f(x)-ax)\end{array}$$

PROOF

TODO