Limits Continuity

MA4001 Engineering Mathematics 1Lecture 10

Limits and Continuity

Dr. Sarah Mitchell

Autumn 2014

Limits Continuity

Infinite limits

If f (x) grows arbitrarily large as x → a we say that f (x) has aninfinite limit.

Example: f (x) =1x2 . lim

x→0f (x) = ∞

The line x = 0 is a vertical asymptote.

Limits Continuity

Example f (x) = 1x

limx→0+

1x= ∞ lim

x→0−

1x= −∞

Therefore limx→0

1x

does not exist.

Limits Continuity

Example f (x) =x2 + 1x2 − 1

Clearly the interesting x values are ±1 and ±∞.

Note that f (−x) = f (x), so the function is even. Thus we onlyneed to check the behaviour at 1 and ∞.

limx→1+

f (x) = +∞ limx→1−

f (x) = −∞ (since x2−1 < 0 for x < 1)

limx→∞

f (x) = limx→∞

1 + 1/x2

1 − 1/x2 = 1

Thus x = ±1 are vertical asymptotes and y = 1 is a horizontalasymptote. Also f (0) = −1.

Limits Continuity

Graph ofx2 + 1x2 − 1

Limits Continuity

Continuity at a point

Definition

f (x) is continuous at c, if

c is an interior point of the domain of f

limx→c

f (x) = f (c) i.e., the graph of f (x) is continuous at c.

If

limx→c

f (x) 6= f (c) or

limx→c

f (x) does not exist

then we say that f (x) is discontinuous at c, i.e., the graph isdiscontinuous at c.

Limits Continuity

Remark

If f (x) is not defined at c, then f (x) is neither continuous nordiscontinuous at c.

Limits Continuity

Example f (x) = x2 + 1

limx→1

f (x) = 12 + 1 = 2 = f (1)

Thus f (x) is continuous at 1.

We could repeat this for any value of x , so in fact f (x) iscontinuous in R.

Limits Continuity

The Heaviside function

H(x) =

{1, x > 0

0, x < 0is continuous in R \ {0}.

H(x) is discontinuous at x = 0 since limx→0

H(x) doesn’t exist.

Limits Continuity

Example: Removable Singularity

f (x) =

{x2 + 1, if x 6= 1

1, if x = 1

f (x) is continuous on R \ {1}.

f (x) is discontinuous at x = 1 since

limx→1

f (x) = limx→1

(x2 + 1) = 12 + 1 = 2 6= f (1) = 1

Limits Continuity

Removable Singularity

Definition

f (x) has a removable singularity at x = c if f (x) can be madecontinuous at c by redefining f (c) to be

f (c) = limx→c

f (x)

In the previous example, f (x) had a removable singularity atx = 1.

By redefining f (1) = 2 the function is made continuous at x = 1.

Limits Continuity

Example sgn(x)

sgn(x) is continuous on its domain R \ {0}.

At x = 0, sgn(x) is not defined. Thus it is neither continuousnor discontinuous at 0.

Limits Continuity

Left and right continuity

Definition

f (x) is right continuous at x = c if limx→c+

f (x) = f (c).

f (x) is left continuous at x = c if limx→c−

f (x) = f (c).

Limits Continuity

Example: Heaviside function

H(x) =

{1, x > 0

0, x < 0is right continuous at x = 0 since

limx→0+

H(x) = 1 = H(0)

but not left continuous at x = 0 since

limx→0−

H(x) = 0 6= 1 = H(0)

Limits Continuity

Continuity at endpoints of domains

Definition

f (x) is continuous at a left endpoint of its domain, if it is rightcontinuous at this point.

f (x) is continuous at a right endpoint of its domain, if it is leftcontinuous at this point.

Limits Continuity

Example f (x) =√

1 − x2

The domain is 1 − x2 > 0, i.e., x2 6 1, i.e., [−1,1]

f (x) is continuous on (−1,1).

f (x) is right continuous at x = −1 and left continuous at x = 1.

Thus f (x) is continuous on [−1,1].

Limits Continuity

Example f (x) =√

x − 2

The domain is x − 2 > 0, i.e., x > 2, i.e., [2,∞)

f (x) is continuous on (2,∞).

f (x) is right continuous at x = 2

Thus f (x) is continuous on its whole domain [2,∞).

Limits Continuity

Continuity on R

Many functions are continuous on R. These are referred to ascontinuous functions.

Examples:

all polynomials;

all rational functions with non-zero denominator;

sin x , cos x , ex

Limits Continuity

Combinations of continuous functions

If f (x) and g(x) are continuous at c then the following arecontinuous at c:

f (x) + g(x)

f (x) − g(x)

f (x)g(x)f(x)g(x) provided g(c) 6= 0.

Limits Continuity

Example

sin x , x , ex are all continuous on R.

Then3 sin x

ex +8x

is continuous on its domain R \ {0}.

Limits Continuity

Continuity of composite functions

(f ◦ g)(x) = f (g(x))

If g(x) is continuous at c and f (x) is continuous at g(c) then(f ◦ g)(x) is continuous at c.

Example. f (x) =√

x , g(x) = x2 − 2x + 5.

(f ◦ g)(x) =√

x2 − 2x + 5 is continuous on R.

Note that the domain of f ◦ g is R.

Limits Continuity

Continuous functions on [a, b]

Recall that f (x) is continuous on the closed interval [a,b] if

f (x) is continuous at each x ∈ (a,b);

f (x) is right continuous at x = a;

f (x) is left continuous at x = b.

Limits Continuity

The max-min theorem

Theorem

If f (x) is continuous on [a,b], there exist x1, x2 ∈ [a,b], such that

f (x1) 6 f (x) 6 f (x2), ∀x ∈ [a,b]

We say that f (x) has the absolute maximum M = f (x2) atx = x2 and the absolute minimum m = f (x1) at x = x1 on [a,b].

Limits Continuity

Corollary

If f (x) is continuous on [a,b], then it is bounded on [a,b]

i.e., there exists K > 0 such that |f (x)| 6 K for all x ∈ [a,b].

Proof.

Choose K = max{|f (x1)|, |f (x2)|}.