2.1 – Rates of Change and Limits

The Epsilon-Delta Definition of a Limit

Given a function $f(x)$, we say that the limit of $f(x)$ as $x$ approaches $c$ is $L$, written as

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

if for every $\epsilon >0$ there exists a $\delta >0$ such that for all $x$,

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

One-Sided Limits

The one-sided limit of a function $f(x)$ is evaluated by approaching a point $a$ exclusively from one side.

Left-sided limit:

$$\underset{x\to a^{-}}{\lim }f(x)$$

Right-sided limit:

$$\underset{x\to a^{+}}{\lim }f(x)$$

Example:

$$f(x)=\{\begin{array}{l}x-5,x\le -2\\ x^2,x>-2\end{array}$$

For the left-sided limit at $x=-2$, ${\lim }_{x\to -2^{-}}f(x)$:

Approaching -2 from the left, $f(x)$ is defined as $x-5$. Substituting -2 for $x$, the left-sided limit at $-2$ is evaluated as $-2-5=-7$.

For the right-sided limit at $x=-2$, $li{m}_{x\to -2^{+}}f(x)$:

Although $f(x)=x-5$ exactly at -2, for values of $x$ just to the right of -2, $f(x)$ is defined as $x^2$. Substituting -2 into $x^2$, we find the right-sided limit at -2 to be 4.

An Analytical Approach

We may find the one-sided limits analytically by using a table of values.

Consider the piecewise function from the previous section:

$$f(x)=\{\begin{array}{l}x-5,x\le -2\\ x^2,x>-2\end{array}$$

$x$	$f(x)$
-2.1	-7.1
-2.01	-7.01
-2.001	-7.001
-2.0001	-7.0001
-2	-7
-1.9999	3.9996…
-1.999	3.996001
-1.99	3.9601
-1.9	3.61

This table illustrates that as $x$ approaches -2 from the left, $f(x)$ approaches -7. As $x$ approaches -2 from the right ($-1.\overline{9}$), $f(x)$ approaches 4.

Note that this does not work for all functions. For example, $\sin (\frac{1}{x})$ oscillates in the neighbourhood of zero, so we do not find it to approach a limit from either side.

Note: Using a table of values does not work for every function to determine the limit. For instance, $\sin (\frac{1}{x})$ oscillates infinitely near zero, making it impossible to determine a limit from either side.

The Overall Limit

For a function to have an “overall” limit at some $c$, the left- and right-sided limits must be equal at $c$.

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

If the two limits are not equal, the limit at $c$ does not exist, which is represented as $\text{DNE}$.

$$\underset{x\to c^{-}}{\lim }f(x)\ne \underset{x\to c^{+}}{\lim }f(x)$$ $$\therefore \underset{x\to c}{\lim }f(x)=\text{DNE}$$

Note that a function does not need to take the value of the limit when it is evaluated at the same point.

For example,

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

However,

$$\frac{(2{)}^2-4}{(2)-2}=\text{undefined}$$

Properties of Limits

Sum	${\lim }_{x\to c}(f(x)+g(x))$	$L+M$
Difference	${\lim }_{x\to c}(f(x)-g(x))$	$L-M$
Product	${\lim }_{x\to c}(k\cdot f(x))$	$(k\cdot L)$
Constant Multiple	${\lim }_{x\to c}(f(x)\cdot g(x))$	$L\cdot M$
Quotient	${\lim }_{x\to c}\frac{f(x)}{g(x)}$	$\frac{L}{M},M\ne 0$
Power	${\lim }_{x\to c}(f(x){)}^{\frac{r}{s}}$	$L^{\frac{r}{s}}$
Constant	${\lim }_{x\to c}k$	$k$

One-Sided

$$\underset{x\to c^{-}}{\lim }f(x)=L⟺\underset{x\to c^{+}}{\lim }=L$$ $$\text{and }\underset{x\to c}{\lim }f(x)=L$$

Average vs. Instantaneous

Rates of Change

$$\bar{v}=\frac{\Delta d}{\Delta t}$$ $$\underset{\Delta t\to 0}{\lim }\frac{\Delta d}{\Delta t}=v_{\text{inst.}}=\frac{dd}{dt}$$

dy and dx

$$\Delta y\to dy$$ $$\Delta x\to dx$$

Limits Using CAS

The limit of an expression can be computed in KhiCas by calling the limit() function, which accepts the following parameters:

• Expr: the mathematical expression to be evaluated
• Var: the variable within Expr to compute the limit of
• Val: the point to which Var approaches
  • Val = -oo: the limit at $-\infty$
  • Val = oo: the limit at $\infty$
• [Dir(d)] (optional): Indicates the direction from which Var approaches Val
  • [Dir(d)] = -1: the left-side limit
  • [Dir(d)] = +1: the right-side limit KhiCas will calculate the two-sided limit if empty

Examples

To compute ${\lim }_{x\to 3}\frac{x^2-9}{x-3}$:

>>> limit((x^2-9)/(x-3), x, 3)
6

To compute ${\lim }_{x\to \infty }\frac{1}{\ln x}$:

>>> limit(1/ln(x), x, oo)
0

Notes

• limit() can be accessed through the calc menu by the following keystrokes:

  • F2 > 4

• $\infty$ is represented by oo (two lowercase o) in KhiCas. The shortcut for oo can be accessed through the algb menu by the following keystrokes:

  • F1 > 7