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(%);