Limits and Continuity
Contents
Limits and Continuity¶
Recall (Function):¶
Definition:¶
A function \(f\) is a rule of correspondence that associates with each element \(x\) in one set \(D\), called the domain, a single value of \(f(x)\) from a second set, \(R\). The set of all values obtained is called the range of the function.
Domain/Range of Functions¶
Let \(y=f(x)\).
Name |
\(f(x)\) |
\(n\in \mathbb{Z}\) |
\(a\in \mathbb{R}\) |
Domain |
Range |
Example |
|---|---|---|---|---|---|---|
Polynomials |
\(f(x)=a_{n}x^{n}+\cdots+a_{1}x+a_{0}\) |
n/a |
n/a |
\(\{ x | x \in (-\infty,\infty) \}\) |
\(\{ y | y \in (-\infty,\infty) \}\) |
\(f(x)=m x+b\) or \(f(x)=ax^2+bx+c\) |
Radical |
\(f(x)=x^{1/n}=\sqrt[n]{x}\) |
\(n\) is even |
n/a |
\(\{ x|x\in[0,\infty) \}\) |
\(\{ y|y\in(-\infty,\infty) \}\) |
\(f(x)=\sqrt{x}\) |
Radical |
\(f(x)=x^{1/n}=\sqrt[n]{x}\) |
\(n\) is odd |
n/a |
\(\{ x|x\in(-\infty,\infty) \}\) |
\(\{ y|y\in(-\infty,\infty) \}\) |
\(f(x)=\sqrt[3]{x}\) |
Reciprocal |
\(f(x)=x^n\) |
\(n<0\) |
n/a |
\(\{ x|x\in(-\infty,0)\cup(0,\infty) \}\) |
\(\{ y|y\in(-\infty,0)\cup(0,\infty) \}\) |
\(f(x)=\frac{1}{x}\) |
Exponential |
\(f(x)=a^x\) |
n/a |
\(a>0\) and \(a\ne 1\) |
\(\{ x|x\in(-\infty,\infty) \}\) |
\(\{ y|y\in (0,\infty)\}\) |
\(f(x)=e^x\) |
Logarithmic |
\(f(x)=\log_a(x)\) |
n/a |
\(a>0\) and \(a\ne 1\) |
\(\{ x|x\in (0,\infty)\}\) |
\(\{ y|y\in(-\infty,\infty) \}\) |
\(f(x)=\log_e(x)=\ln(x)\) |
Combination of Functions:¶
Let \(f\) and \(g\) be two functions.
\(\left( f\pm g \right)(x)=f(x)\pm g(x)\)
\(\left( f\cdot g \right)(x)=f(x)\cdot g(x)\)
\(\left( \frac{f}{g} \right)(x)=\left(f\div g\right)(x)=f(x)\div g(x)=\dfrac{f(x)}{g(x)}\) where \(g(x)\ne 0\) everywhere.
\(\left( f\circ g \right)(x)=f(g(x))\) the \(\circ\) is the composition of functions operator.
Notice that \(f\circ g \ne g\circ f\).
Monotonicity of Functions:¶
Let \(f\) be a function of \(x\) and for every \(x_1\) and \(x_2\), \(a\le x_1<x_2 \le b\):
If \(f(x_1)<f(x_2)\), then \(f\) is called an increasing function on the interval \([a,b]\).
If \(f(x_1)>f(x_2)\), then \(f\) is called a decreasing function on the interval \([a,b]\).
Recall (Limits):¶
Suppose
Then
\({\displaystyle \lim_{x\to a}\left[f(x)\right]}^{r}{\displaystyle =\left[\lim_{x\to a}f(x)\right]^{r}=L^{r}}\) where \(r\) is a positive number.
\({\displaystyle \lim_{x\to a}cf(x)=c\lim_{x\to a}f(x)=cL}\) where \(c\) is a real number.
\({\displaystyle \lim_{x\to a}\left[f(x)\pm g(x)\right]=\lim_{x\to a}f(x)\pm\lim_{x\to a}g(x)=L\pm M}\).
\({\displaystyle \lim_{x\to a}\left[f(x)g(x)\right]=\left[\lim_{x\to a}f(x)\right]\left[\lim_{x\to a}g(x)\right]=L\cdot M}\).
\({\displaystyle \lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}=\frac{L}{M}}\) where \(M\ne0\).
\({\displaystyle \lim_{x\to a}x}=a\)
\({\displaystyle \lim_{x\to a}b}=b\)
\({\displaystyle \lim_{x\to \infty}\frac{1}{x}}=0\) (thm. 5)
If \(f(x)=g(x)\) for all \(x\ne a\), then \({\displaystyle \lim_{x\to a}f(x)=\lim_{x\to a}g(x)}\). (cor. 1)
If \(\lim_{x\to a^{+}}f(x)\ne\lim_{x\to a^{-}}f(x)\), then we say \(\lim_{x\to a}f(x)\) does not exists. (cor. 2)
Continuity of Functions¶
Definition (Three steps to continuity):¶
A function \(f\) is continuous at a number \(x=a\) if the following conditions are satisfied.
\(f(x)\) is defined
\({\displaystyle \lim_{x\to a}f(x)}\) exists
\({\displaystyle \lim_{x\to a}f(x)=f(a)}\)
If \(f\) is not continuous at \(x=a\), then \(f\) is said to be discontinuous at \(x=a\).
Properties of Continuous Functions:¶
The constant function \(f(x)=c\) is continuous everywhere.
The identity function \(f(x)=x\) is continuous everywhere.
If \(f\) and \(g\) are continuous at \(x=a\), then
\(f\pm g\) is continuous at \(x=a\).
\(fg\) is continuous at \(x=a\).
\(\frac{f}{g}\) is continuous at \(x=a\) provided that \(g(a)\ne0\).
Let \(P\) be a polynomial function and \(R\) be a rational function (\(R(x)=\frac{p(x)}{q(x)}\) where \(p\) and \(q\) are polynomials), then
\(P\) is continuous everywhere.
\(R\) is continuous everywhere except when \(q(x)=0.\)
Limits and Continuous Functions:¶
Let \(p(x)\) and \(q(x)\) be polynomial functions. Let \(a\) be a real number. Then: 1.
Intermediate Value Theorem (IVT)¶
If \(f\) is a continuous function on a closed interval \([a,b]\), and if \(y_{0}\) is any value between \(f(a)\) and \(f(b)\), then \(y_{0}=f(c)\) for some \(c\) in \([a,b]\).
Definition¶
We say the solution of th equation \(f(x)=0\) is a root of the equation or the zero of the function \(f\).
Theorem¶
Let \(f\) be continuous on the interval \([a,b]\). Then the following are true:
If \(f(x)>0\) for all \(x\in[a,b]\) then \(f\) has no roots for all \(x\in[a,b]\).
If \(f(x)<0\) for all \(x\in[a,b]\) then \(f\) has no roots for all \(x\in[a,b]\).
Corollary¶
Let \(f\) be continuous on the interval \([a,b]\). Then the following are true:
If there exists a \(c\in(a,b)\) such that \(f(c)=0\) and \(f(a)<0\), then \(f(b)>0\).
If there exists a \(c\in(a,b)\) such that \(f(c)=0\) and \(f(a)>0\), then \(f(b)<0\).
Corollary¶
Let \(f\) be continuous on the interval \((a,b)\). Let \(a<x_1<c<x_2<b\) and \(f(c)=0\). Then the following are true:
If \(f(x_1)<0\), then \(f(x)\le 0\) for all \(x\in(a,c]\).
If \(f(x_1)>0\), then \(f(x)\ge 0\) for all \(x\in(a,c]\).
If \(f\) is continuous on the interval \((a,b)\), there exists a \(c\in(a,b)\) such that \(a<x_1<c<x_2<b\) and \(f(c)=0\), and \(f(x_2)>0\), then what can be said about all \(x\in[c,b)\)?
Example¶
Let \(f(x)=x^2-1\). Where is \(f(x)\) positive and where is \(f(x)\) negative.
SOLUTION:
Notice that \(f(1)=0\), \(f(-1)=0\), and \(f\) is continuous on \((-\infty,\infty)\).
Choose \(x_1=-2<-1\). Since \(f(-2)=3\) we have by Corollary that \(f(x)>0\) for all \(x\in(-\infty,-1)\).
Choose \(x_1=0<1\). Since \(f(0)=-1\) we have by Corollary that \(f(x)<0\) for all \(x\in(-1,1)\).
Choose \(x_1=2>1\). Since \(f(2)=3\) we have by Corollary that \(f(x)>0\) for all \(x\in(1,\infty)\).
Finally, we conclude that
\(f(x)>0\) for all \(x\in(-\infty,-1)\cup(1,\infty)\) and
\(f(x)<0\) for all \(x\in(-1,1)\).
Consider the graph:
plot2d(x^2-1,[x,-3,3]);
Exercise:¶
Let \(f(x)=\frac{x^3}{3}-4 x\). Find where \(f\) is positive or negative using the same reasoning above. The solution should follow the image below:
[HINT
Solve \(f(x)=0\).
Create 4 subintervals.
Choose \(x_1\) in each of the subintervals.
Make observations based on the Corollary.]
plot2d(x^3/3-4*x,[x,-5,5]);
Example¶
Let
Find \(k\) such that \(f\) is continuous at \(x=1\).
SOLUTION:
Need to find \(k\) such that:
\(\lim_{x\to 1}f(x)\) exists.
\(f(1)\) exists.
\(\lim_{x\to1}f(x)=f(1)\).
First notice:
(1) Evaluate
which means \(\lim_{x\to1}f(x)\) exists.
(2) Evaluate
which means \(f(1)\) exists.
(3) In order for \(\lim_{x\to1}f(x)=f(1)\) we must set \(k=2\). If \(k=2\), then
That is, in order for \(f\) to be continuous at \(x=1\), \(k\) must be \(2\).
Example¶
Let
Find \(k\) such that \(f\) is continuous at \(x=a\).
SOLUTION:
Need to find \(k\) such that:
\(\lim_{x\to a}f(x)\) exists.
\(f(a)\) exists.
\(\lim_{x\to a}f(x)=f(a)\).
First notice:
(1) First, simplify
Evaluate
which means \(\lim_{x\to a}f(x)\) exists.
(2) Evaluate
which means \(f(a)\) exists.
(3) In order for \(\lim_{x\to a}f(x)=f(a)\) we must set \(k=2a\). If \(k=2a\), then
That is, in order for \(f\) to be continuous at \(x=a\), \(k\) must be \(2a\).
Exercise¶
Recall that the average rate of change of a function \(f\) is
Let \(f(x)=x^2\).
Find the average rate of change of the function \(f\) as \(x\) goes from \(1\) to \(2\).
Find the average rate of change of the function \(f\) as \(x\) goes from \(0\) to \(1\).
Find the average rate of change of the function \(f\) as \(x\) goes from \(0.5\) to \(1\).
Find the average rate of change of the function \(f\) as \(x\) goes from \(1\) to \(1.5\).
Attempt to find the instantaneous rate of change of the function \(f\) at \(x=1\).
(For (5.) use the method worked out from the previous example; here is the solutions to each part.)
f(x):=x^2$
m(x1,x2):=(f(x2)-f(x1))/(x2-x1)$
m(2,1);
m(0,1);
m(0.5,1); m(1,1.5);
limit(m(x,1),x,1);