Unit 1: Limits and Continuity

**Limits**

Limits are the value that a function approaches as the variable within the function gets nearer to a particular value.

We don’t really care what’s happening

**at**the point, we care about what’s happening**around**the pointTo find the limit of a simple polynomial, plug in the number that the variable is approaching

### Ways to Find Limits

Look on a graph to see what it approaches

If the graph approaches two different values for the same number, the limit does not exist

Estimate from a table

Algebraic Properties

Algebraic Manipulation

You can factor the numerator and denominator, then cancel any removable discontinuities

This is mostly useful if you get limits where the denominator is equal to 0

For example, (x+3)(x+2)/(x+3)(x-3)

(x+3) is able to be removed → removable discontinuity

### Squeeze Theorem

Conditions

For all values of

*x*in the interval that contains*a*,*g(x) ≤ f(x) ≤ h(x)**g*and*h*have the same limit as*x*approaches*a*

lim g(x) = L, lim h(x) = L, therefore lim f(x) = L

Trig limits as x approaches 0:

lim [sin(x)/x] = 1

lim [(cos(x)-1)/x] = 0

lim [sin(ax)/x] = a

lim [sin(ax)/sin(bx)] = a/b

**Continuity**

**Jump Discontinuity**Occurs when the curve “breaks” at a particular place and starts somewhere else

The limits from the left and the right will both exist, but they will not match

**Essential/Infinite Discontinuity**The curve has a vertical asymptote

**Removable Discontinuity**An otherwise continuous curve has a hole in it

“Removable” because one can remove the discontinuity by filling the hole

**Continuity Conditions**For f(x) to be continuous when x=c:

f(c) exists

the limit as x→c exists

lim f(x) = f(c)

x→c

A function is continuous on an interval if it is continuous at every point on that interval

**Removing Discontinuities**

You can remove a discontinuity by redefining the function without that point in the domain

This is frequently done by factoring out a common root between the numerator and denominator

**Limits and Asymptotes**

Vertical asymptote: a line that a function cannot cross because the function is undefined there

Horizontal asymptote: the end behavior of a function

A horizontal asymptote can be crossed

Horizontal Asymptote Rules

If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is infinity: there is no horizontal asymptote

If the highest power of x is in the denominator, then the limit as x approaches infinity is zero and the horizontal asymptote is the line

*y=0*If the highest power is the same, then the limit is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator

Intermediate Value Theorem

Guarantees that if a function f(x) is continuous on the interval [a,b] and C is any number between f(a) and f(b), ten there is at least one number in the interval [a,b] such that f(x) = C

# Unit 1: Limits and Continuity

**Limits**

Limits are the value that a function approaches as the variable within the function gets nearer to a particular value.

We don’t really care what’s happening

**at**the point, we care about what’s happening**around**the pointTo find the limit of a simple polynomial, plug in the number that the variable is approaching

### Ways to Find Limits

Look on a graph to see what it approaches

If the graph approaches two different values for the same number, the limit does not exist

Estimate from a table

Algebraic Properties

Algebraic Manipulation

You can factor the numerator and denominator, then cancel any removable discontinuities

This is mostly useful if you get limits where the denominator is equal to 0

For example, (x+3)(x+2)/(x+3)(x-3)

(x+3) is able to be removed → removable discontinuity

### Squeeze Theorem

Conditions

For all values of

*x*in the interval that contains*a*,*g(x) ≤ f(x) ≤ h(x)**g*and*h*have the same limit as*x*approaches*a*

lim g(x) = L, lim h(x) = L, therefore lim f(x) = L

Trig limits as x approaches 0:

lim [sin(x)/x] = 1

lim [(cos(x)-1)/x] = 0

lim [sin(ax)/x] = a

lim [sin(ax)/sin(bx)] = a/b

**Continuity**

**Jump Discontinuity**Occurs when the curve “breaks” at a particular place and starts somewhere else

The limits from the left and the right will both exist, but they will not match

**Essential/Infinite Discontinuity**The curve has a vertical asymptote

**Removable Discontinuity**An otherwise continuous curve has a hole in it

“Removable” because one can remove the discontinuity by filling the hole

**Continuity Conditions**For f(x) to be continuous when x=c:

f(c) exists

the limit as x→c exists

lim f(x) = f(c)

x→c

A function is continuous on an interval if it is continuous at every point on that interval

**Removing Discontinuities**

You can remove a discontinuity by redefining the function without that point in the domain

This is frequently done by factoring out a common root between the numerator and denominator

**Limits and Asymptotes**

Vertical asymptote: a line that a function cannot cross because the function is undefined there

Horizontal asymptote: the end behavior of a function

A horizontal asymptote can be crossed

Horizontal Asymptote Rules

If the highest power of x in a rational expression is in the numerator, then the limit as x approaches infinity is infinity: there is no horizontal asymptote

If the highest power of x is in the denominator, then the limit as x approaches infinity is zero and the horizontal asymptote is the line

*y=0*If the highest power is the same, then the limit is the coefficient of the highest term in the numerator divided by the coefficient of the highest term in the denominator

Intermediate Value Theorem

Guarantees that if a function f(x) is continuous on the interval [a,b] and C is any number between f(a) and f(b), ten there is at least one number in the interval [a,b] such that f(x) = C