L3-continuity2.mws

__Calculus I__

**Lesson 3: Continuity and Limits at Infinity II**

**Example 1**

Plot
. Determine the limit of f(x) as x goes to
.

`> `
**restart: **

`> `
**a:= x * sin(1/x) ;**

`> `
**plot(a(x), x = 100..1000);**

From the graph, we conjecture that the limit of f(x) as x goes to
is 1.

**Example 1b**

Plot
. Determine the limit of g(x) as x goes to
.

`> `
**b:= 2^x / x;**

`> `
**plot(b(x), x = 10..100);**

From the graph we conjecture that the limit of g(x) as x goes to
is
.

**Example 2**

Plot
. Is this function continuous at 0? Does the limit of

this function exist as x goes to zero? What about the limit as x goes to
?

Where is the function continuous?

`> `
**c:= (1/x)*sin(1/x);**

`> `
**plot(c(x), x = 0.01..1);**

`> `
**plot(c(x), x = -0.01..-1);**

`> `
**plot(c(x), x = 100..1000);**

**Conclusions**
: The function is NOT continuous at x = 0 since the limit

of the function as x goes to zero does not exist. However, the limit of

the function as x goes to
is 0. The function is continuous everywhere

except at x = 0.

**Example 3**

a) Let
and
. Graph s(x) and t(x) on the same

axes with different colors.

b) Suppose that s(x) < f(x) < t(x) for all x. Find the limit of f(x) as x goes to 4.

`> `
**s:= x^2 - 4*x;**

`> `
**t:= 4*x - x^2;**

`> `
**plot([s(x),t(x)], x= 2..6, color=[magenta, brown]);**

*answer to (3b): limit of f(x) as x goes to 4 is zero. *

**Example 4 **

a) Let
and
. Plot u(x) and v(x) on

the same axes with different color.

b) Suppose that u(x) < f(x) < v(x) for all x > 5. Find limit of f(x) as x goes to infinity.

`> `
**u:= (4*x - 1)/x;**

`> `
**v(x):= (4*x^2 + 3*x)/(x^2);**

`> `
**plot([u(x),v(x)],x = 10..100, color=[magenta,brown]);**

`> `
**plot([u(x),v(x)],x = 1000..10000, color=[magenta,brown]);**

a
*nswer to (4b) : From the graph, limit of f(x) as x goes to infinity is 4.*

* *

**Example 5**

Let
. Plot m(x) to obtain the limit

of m(x) as x goes to negative infinity. Compute the limit without using Maple.

`> `
**m:= (sqrt(4*x^2 + 1)) /(x + 1);**

`> `
**plot(m(x), x = -10..-100);**

`> `
**plot(m(x), x = -100..-1000);**

`> `
**limit(m(x), x = -infinity);**

`> `
**evalf(%);**