Limits

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible. 

the formal $\delta -\in$definition of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L,\underset{x\to c^{-}}{\lim }f\left(x\right)=L$, and $\underset{x\to c^{+}}{\lim }f\left(x\right)=L$

Fill in the blanks to complete each of the following theorem statements: 

For $\delta >0, x\in (c-\delta ,c)\cup (c,c+\delta )$if and only if $0<\underline{ }<\delta .$

what we mean when we say that a limit exists, or that a limit does not exist.

Fill in the blanks to complete each of the following theorem statements: 

The Extreme Value Theorem: If f is on a closed interval$[a,b]$, then there exist values M and m in the interval$[a,b]$ such that $f\left(M\right)$is        and $f\left(m\right)$is         . 

what it means, in terms of limits, for a function f to have a vertical asymptote at x = c or a horizontal asymptote at y = L 

Fill in the blanks to complete each of the following theorem statements: 

The Intermediate Value Theorem: If f is          on a closed interval$[a,b]$, then for any K strictly between          and         , there exists at least one $c\in (a,b)$such that         . 

what it means, in terms of limits, for a function f to be continuous at a point$x=c$, left continuous at$x=c$, and right continuous at$x=c.$ 

Fill in the blanks to complete each of the following theorem statements: 

A function f can change sign from positive to negative, or vice versa, at$x=c$ only if $f\left(x\right)$is         ,         , or         at$x=c$.

what it means for a function f to be continuous on a closed interval [a, b], or continuous on an open interval (a, b), or continuous on a half-closed interval [a, b) 

Fill in the blanks to complete each of the following theorem statements: 

Constant, identity, and linear functions are continuous everywhere, which means in terms of limits that        ,         , and           . 

Fill in the blanks to complete each of the following theorem statements: 

Power functions are continuous everywhere, which means in terms of limits that             . 

the definition of the number e in terms of a limit.

Fill in the blanks to complete each of the following theorem statements: 

All algebraic functions are continuous on their domains, which means in terms of limits that if $x=c$is in the domain of an algebraic function f, then            . 

All basic transcendental functions are ? on their domains, which means in terms of limits that if $x=c$ is in the domain of a basic exponential, logarithmic, trigonometric, or inverse trigonometric function $f$ , then ?. 

Fill in the blanks to complete each of the following theorem statements: 

If $\underset{x\to c}{\lim }g\left(x\right)$ exists and $f\left(x\right)$ is a function that is ? to $g\left(x\right)$ for all $x$ sufficiently close to ?, but not necessarily at ? , then  ?.

The Squeeze Theorem for Limits: If $I\left(x\right)\le f\left(x\right)\le u\left(x\right)$ for all $x$sufficiently close to ?, but not necessarily at ?, and if $\underset{x\to c}{\lim }l\left(x\right)\text{ and }\underset{x\to c}{\lim }u\left(x\right)$are both equal to L, then ?. 

Suppose$\frac{p\left(x\right)}{q\left(x\right)}$ is a rational function with deg$\left(p\right(x\left)\right)$$=n$and$deg\left(q\right(x\left)\right)$$=m,$if $n<m$then,$\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?,$ if $n=m,$ then $\underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?,$and if   $n>m, then \underset{x\to \infty }{\lim }\frac{p\left(x\right)}{q\left(x\right)}=?.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

$\underset{x\to c}{\lim }k=?.$

Calculating limits: Find each limit by hand.

$\underset{x\to 0}{\lim }3x^{-4}.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

$\underset{x\to c}{\lim }x=?.$

Calculating limits: Find each limit by hand.

$\underset{x\to \infty }{\lim }-2x^{-1/2}.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to c}{\lim }(mx+b)=?$

Calculating limits: Find each limit by hand.

$\underset{x\to 2}{\lim }\frac{1}{2-x}.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to c}{\lim }A{x}^n=?.$

Calculating limits: Find each limit by hand.

$\underset{x\to 1}{\lim }\frac{1}{x^2-1}.$

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to \infty }{\lim }{x}^k=?$

Calculating limits: Find each limit by hand.

$\underset{x\to 1}{\lim }\frac{2x^3-x^2-2x+1}{x^2-2x+1}$.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive .

$\underset{x\to \infty }{\lim }{x}^{-k}=?.$

Calculating limits: Find each limit by hand.

$\underset{x\to -\infty }{\lim }\frac{x^3+2x+1}{1-x^4}.$

Consider the limit expression

$\underset{x\to 0}{\lim }\frac{3^x-4^x}{3^x}$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to c}{\lim }{b}^x=?.$

Consider the limit expression 

$\underset{x\to 0}{\lim }\frac{e^x-1}{3e^{2x}-2e^x-1}$

Find the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to c}{\lim }\sin x$= _

Consider the limit expression 

$\underset{x\to 0^{+}}{\lim }\frac{\ln x}{x}$

Calculate the limit. 

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$is positive 

$\underset{x\to \infty }{\lim }{e}^{kx}=?.$

Consider the limit expression

$\underset{x\to \infty }{\lim }\ln \left(\frac{x-1}{1-3x^2}\right)$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$is positive 

$\underset{x\to \infty }{\lim }{e}^{-kx}=?$

Consider the limit expression

$\underset{x\to \infty }{\lim }\frac{1-e^x}{e^{2x}}$

Calculate the limit

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive 

$\underset{x\to \infty }{\lim }{2}^x$=?.

Consider the limit expression

$\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin x}{x}$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$ is positive 

$\underset{x\to \infty }{\lim }(0.75{)}^x$=?.

Consider the limit expression

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

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$ is positive. 

$\underset{x\to 0^{+}}{\lim }\ln x=?$

Consider the limit expression

$\underset{x\to 0^{+}}{\lim }\frac{\sqrt{x}-x^3}{x^2}$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$ is positive.

$\underset{x\to \infty }{\lim }\ln x=-.$

$\underset{x\to 3}{\lim }\frac{\frac{1}{x-3}-\frac{1}{x}}{x-3}$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$ is positive. 

$\underset{x\to \infty }{\lim }{\tan }^{-1}x=-.$

$\underset{x\to 4}{\lim }\frac{2-\sqrt{x}}{4-x}$

Calculate the limit.

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that $k$ is positive.

$\underset{x\to -\infty }{\lim }{\tan }^{-1}x=-.$

$\underset{x\to \infty }{\lim }\left(-2x^3+x^2-10\right)$

Calculate the limit.