 Application Center - Maplesoft

# Continuity and limits at infinity I

You can switch back to the summary page by clicking here.

L2-continuity1.mws

Calculus I

Lesson 2: Continuity and Limits at Infinity I

Example 1
For the following function, find the value of
a that makes the function continuous.

Plot the continuous function. Then take different values of the variable a and

plot the associated discontinuous functions. for x <= 2 and for x > 2

> restart:

Define the two pieces

> f1 := x->a*x; f2 := x->a*x^2+x+1;  Substitute the break point x=2 into both functions, set them equal, and then solve for a:

> a := solve(f1(2)=f2(2), a); > f1(x); f2(x);  Plot the function.

> f:= x -> piecewise( x <= 2, f1(x), f2(x)); > plot(f(x), x = -5..5); What if we'd gotten a wrong? Suppose we thought a = -4.

> a := -4; > f1(x); f2(x);  Plot the (discontinuous) function.

> f:= x -> piecewise( x <= 2, f1(x), f2(x)); > plot(f(x), x = -5..5, discont=true); Example 2

Plot the graph of the following function.

Where is the function discontinuous?

q(t) = 0 for t < -4

= for -4 <= t <= -2

= for -2 < t <= 1

= for 1 < t <= 6

= 7 for t > 6

> c:= x -> piecewise( x < -4, 0, -4 <= x and x <= -2, 4*x + 16, -2 < x and x <= 1, 2*x^2, 1 < x and x <= 6, 5 * sqrt(x + 3) - 8, 7); > plot(c(x),x= -5..7, discont=true, color=magenta); answer: Function is continuous everywhere; no discontinuity.

Example 3
For each of the following functions:

(i) Plot each function for large vlaues of x.

(ii) Use the plot to conjecture the limitof the function as x goes to (iii) Determine precisely the limit of the function as x goes to .

a) b) c) > e:= ( ( 1+x)/x )* sin(x); > plot(e(x), x = 100..200); > limit((( 1+x)/x)* sin(x), x = infinity); limit of e(x) as x goes to DNE, since the function oscillates between 1 and -1.

> f:= ( 1/(sqrt(x)) * sin(x)); > plot(f(x), x = 1000..2000, numpoints=1000); > limit(( 1/(sqrt(x)) * sin(x)), x = infinity); > g:= (2*x^3 + 7*x)/(x^2); > plot(g(x), x= 1000..2000); > limit((2*x^3 + 7*x)/(x^2), x = infinity); Example 4
A hot piece of steel which is 94 degrees C is placed in a room that is kept at a constant temperature

of 20 degrees C. The steel begins to cool, and has its temperature t minutes after being placed in

the room given by: .

a) Graph T(t)

b) How long till the steel reaches 22 degrees?

c) How long till the steel reaches 21 degrees?

d) What will be the ultimate temp of the steel?

> T:= 20 + 74 * exp( ln(.4)*x/40 ); > plot(T(x), x =0..200 ); > limit(20 + 74 * exp( ln(.4)*x)/40 ), x = infinity); > eq:= 20 + 74 * exp(( (ln(.4)*x)/40 ) )= 22; > solve(eq,x); > eq:= 20 + 74 * exp(( (ln(.4)*x)/40 ) )= 21; > solve(eq); It takes approximately 157 minutes to reach 22 F and 187 minutes to reach 21F. The limiting

temp is 20F.

Example 5
Use Maple's limit command to find each of the following limits.

a) b) c) d) e) > limit( (3 ^x -1)/x, x = 0); > limit( ( 2 - 2^x)/(4^x - 4), x = 1); > limit((1 - cos(x))/(x^2),x = 0); > limit( (x - sin(x))/(x^3), x = 0); > limit( ( 1 + x)^(1/x), x = 0); Example 6
Plot each of the following:

a) b) In each case, plot the asymptotic curves on the same axes with different color.

> plot( [(x^3 - 3)/x, x^2], x= 1..10, color=[green,magenta]); Asymptotic curve is > plot([ (1 + 2*x - x^4)/(x^2), -x^2], x = 1..10, color=[green, magenta]); Example 7

f(x) = x -3 for x > 0

= 5 for x = 0

= for x < 0

Plot f(f(x)) and find limit f(f(x)) as x goes to 0 with ` limit ' command.

> z:= x -> piecewise( x > 0, x-3, x = 0, 5, x^2 + 4*x - 1); > plot(z(x), x = -5..5, discont=true, color=brown); > plot(z(z(x)), x = -5..5, discont=true, color=brown); > limit(z(z(x)), x = 0); Example 8
Find all asymptotes for . Graph f(x) and

all asymptotes on the same axes with different color.

> plot([(3*x^3 - 4*x^2 - 2*x + 5)/(x^2 - 1), 3*x - 4], x = -4..4, color=[red, green], thickness=2); > plot([(3*x^3 - 4*x^2 - 2*x + 5)/(x^2 - 1), 3*x - 4], x = 1.5..10, color=[red, green], thickness=2); 