A Collection of Problems in Differential Calculus: Problems from Calculus I Final Examinations, 2000-2020 Department of Mathematics, Simon Fraser University

Section 1.1 Introduction

Limit

We write $\displaystyle \lim _{x\to a}f(x)=L$ and say “the limit of $f(x)\text{,}$ as $x$ approaches $a\text{,}$ equals $L$” if it is possible to make the values of $f(x)$ arbitrarily close to $L$ by taking $x$ to be sufficiently close to $a\text{.}$

Limit - $\mathbf{\varepsilon,\ \delta}$ Definition

Let $f$ be a function defined on some open interval that contains $a\text{,}$ except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L\text{,}$ and we write $\displaystyle \lim _{x\to a}f(x)=L$ if for every number $\varepsilon >0$ there is a $\delta >0$ such that $|f(x)-L|\lt \varepsilon$ whenever $0\lt |x-a|\lt \delta\text{.}$

Limit and Right-hand and Left-hand Limits

$\displaystyle \displaystyle \lim _{x\to a}f(x)=L\Leftrightarrow (\lim _{x\to a^-}f(x)=L\mbox{ and } \lim _{x\to a^+}f(x)=L)$

Infinite Limit

Let $f$ be a function defined on a neighbourhood of $a\text{,}$ except possibly at $a$ itself. Then $\displaystyle \lim _{x\to a}f(x)=\infty$ means that the values of $f(x)$ can be made arbitrarily large by taking $x$ sufficiently close to $a\text{,}$ but not equal to $a\text{.}$

Vertical Asymptote

The line $x=a$ is called a vertical asymptote of the curve $y=f(x)$ if at least one of the following statements is true:

$$\begin{equation*} \begin{array}{lll} \displaystyle \lim _{x\to a}f(x)=\infty \amp \displaystyle \lim _{x\to a^-}f(x)=\displaystyle \infty \amp \displaystyle \lim _{x\to a^+}f(x)=\infty \\ \displaystyle \lim _{x\to a}f(x)=-\infty \amp \displaystyle \lim _{x\to a^-}f(x)=-\infty \amp \displaystyle \lim _{x\to a^+}f(x)=-\infty \end{array} \end{equation*}$$

Limit At Infinity

Let $f$ be a function defined on $(a,\infty )\text{.}$ Then $\displaystyle \lim _{x\to \infty }f(x)=L$ means that the values of $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently large.

Horizontal Asymptote

The line $y=a$ is called a horizontal asymptote of the curve $y=f(x)$ if if at least one of the following statements is true:

$$\begin{equation*} \lim _{x\to \infty }f(x)=a \mbox{ or } \lim _{x\to -\infty }f(x)=a\text{.} \end{equation*}$$

Limit Laws

Let $c$ be a constant and let the limits $\displaystyle \lim _{x\to a}f(x)$ and $\displaystyle \lim _{x\to a}g(x)$ exist. Then

1. $\displaystyle \displaystyle \lim _{x\to a}(f(x)\pm g(x))=\lim _{x\to a}f(x)\pm\lim _{x\to a}g(x)$

2. $\displaystyle \displaystyle \lim _{x\to a}(c\cdot f(x))=c\cdot \lim _{x\to a}f(x)$

3. $\displaystyle \displaystyle \lim _{x\to a}(f(x)\cdot g(x))=\lim _{x\to a}f(x)\cdot \lim _{x\to a}g(x)$

4. $\displaystyle \lim _{x\to a}\frac{f(x)}{g(x)}=\frac{\lim _{x\to a}f(x)}{\lim _{x\to a}g(x)}$ if $\lim _{x\to a}g(x)\not= 0\text{.}$

Squeeze Law

If $f(x)\leq g(x)\leq h(x)$ when $x$ is near $a$ (except possibly at $a$) and $\displaystyle \lim _{x\to a}f(x)=\lim _{x\to a}h(x)=L$ then $\displaystyle \lim _{x\to a}g(x)=L\text{.}$

Trigonometric Limits

$\displaystyle \lim_{\theta \to 0}\frac{\sin{\theta}}{\theta}=1$ and $\displaystyle \lim_{\theta \to 0}\frac{\cos{\theta}-1}{\theta}=0\text{.}$

The Number $\mathbf{e}$

$\displaystyle \lim_{x \to 0}(1+x)^{\frac{1}{x}}=e$ and $\displaystyle \lim_{x \to \infty }\left( 1+\frac{1}{x}\right) ^x=e\text{.}$

L'Hospital's Rule

Suppose that $f$ and $g$ are differentiable and $g'(x)\not= 0$ near $a$ (except possibly at $a\text{.}$) Suppose that $\ds \lim _{x\to a}f(x)=0$ and $\ds \lim _{x\to a}g(x)=0$ or that $\ds \lim _{x\to a}f(x)=\pm \infty$ and $\ds \lim _{x\to a}g(x)=\pm \infty\text{.}$ Then $\ds \lim _{x\to a}\frac{f(x)}{g(x)}=\lim _{x\to a}\frac{f'(x)}{g'(x)}$ if the limit on the right side exists (or is $\infty$ or $-\infty$).

Continuity

We say that a function $f$ is continuous at a number $a$ if $\displaystyle \lim _{x\to a}f(x)=f(a)\text{.}$

Continuity and Limit

If $f$ is continuous at $b$ and $\displaystyle \lim _{x\to a}g(x)=b$ then $\displaystyle \lim _{x\to a}f(g(x))=f(\lim _{x\to a}g(x))=f(b)\text{.}$

Intermediate Value Theorem

Let $f$ be continuous on the closed interval $[a,b]$ and let $f(a)\not= f(b)\text{.}$ For any number $M$ between $f(a)$ and $f(b)$ there exists a number $c$ in $(a,b)$ such that $f(c)=M\text{.}$