# 1.2 Limits Algebraicallyap Calculus

- 1.2 Limits Algebraicallyap Calculus 2nd Edition
- 1.2 Limits Algebraicallyap Calculus Frq
- 1.2 Limits Algebraicallyap Calculus 14th Edition
- 1.2 Limits Algebraicallyap Calculus Calculator

1.2 Limits Algebraically Remote Checklist (This takes 2 days) Take notes from video; Complete Hw; Notes 1.2 Key. Find Limits of Functions in Calculus. As x gets larger, the terms 1/x and 1/x 2 approach zero and the limit is = 1 / 2. Example 15 Find the limit Solution to.

##### Theorem1.3.1Basic Limit Properties

Let (btext{,}) (ctext{,}) (L) and (K) be real numbers, let (n) be a positive integer, and let (f) and (g) be functions with the following limits: begin{align*}lim_{xto c}f(x)amp=Lamplim_{xto c} g(x)amp = Ktext{.}end{align*}

The following limits hold.

(limlimits_{xto c} b = b)

(limlimits_{xto c} x = c)

(limlimits_{xto c}(f(x)pm g(x)) = Lpm K)

(limlimits_{xto c}(bcdot f(x)) = bL)

(limlimits_{xto c} (f(x)cdot g(x)) = LK)

(limlimits_{xto c} (f(x)/g(x)) = L/Ktext{,}) when (Kneq 0)

(limlimits_{xto c} f(x)^n = L^n)

(limlimits_{xto c} sqrt[n]{f(x)} = sqrt[n]{L})

(If (n) is even, (L) must be non-negative.)

If either of the following holds:

(limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (g(L)=K)

(limlimits_{xto c}f(x)=Ltext{,}) (limlimits_{xto L}g(x)=Ktext{,}) and (f(x)neq L) for all (x) close to but not equal to (c)

then (limlimits_{xto c}g(f(x)) = Ktext{.})

## Approaching ..

Sometimes we can't work something out directly .. but we**can**see what it should be as we get closer and closer!

### Example:

*(x ^{2} − 1)*

**(x − 1)**

Let's work it out for x=1:

*(1 ^{2 }− 1)*

**(1 − 1)**=

*(1 − 1)*

**(1 − 1)**=

*0*

**0**

Now 0/0 is a difficulty! We don't really know the value of 0/0 (it is 'indeterminate'), so we need another way of answering this.

So instead of trying to work it out for x=1 let's try **approaching** it closer and closer:

### Example Continued:

x | (x^{2} − 1)(x − 1) |

0.5 | 1.50000 |

0.9 | 1.90000 |

0.99 | 1.99000 |

0.999 | 1.99900 |

0.9999 | 1.99990 |

0.99999 | 1.99999 |

.. | .. |

Now we see that as x gets close to 1, then *(x ^{2}−1)*

**(x−1)**gets

**close to 2**

We are now faced with an interesting situation:

- When x=1 we don't know the answer (it is
**indeterminate**) - But we can see that it is
**going to be 2**

We want to give the answer '2' but can't, so instead mathematicians say exactly what is going on by using the special word 'limit'.

The **limit** of *(x ^{2}−1)*

**(x−1)**as x approaches 1 is

**2**

And it is written in symbols as:

*lim***x→1***x ^{2}−1*

**x−1**= 2

So it is a special way of saying,* 'ignoring what happens when we get there, but as we get closer and closer the answer gets closer and closer to 2'*

As a graph it looks like this: So, in truth, we But we |

## Test Both Sides!

It is like running up a hill and then finding the path** is magically 'not there'..**

.. but if we only check one side, who knows what happens?

So we need to test it **from both directions** to be sure where it 'should be'!

### Example Continued

So, let's try from the other side:

x | (x^{2} − 1)(x − 1) |

1.5 | 2.50000 |

1.1 | 2.10000 |

1.01 | 2.01000 |

1.001 | 2.00100 |

1.0001 | 2.00010 |

1.00001 | 2.00001 |

.. | .. |

Also heading for 2, so that's OK

## When it is different from different sides

How about a function **f(x)** with a 'break' in it like this:

The limit does not exist at 'a'

**We can't say what the value at 'a' is**, because there are two competing answers:

- 3.8 from the left, and
- 1.3 from the right

But we **can** use the special '−' or '+' signs (as shown) to define one sided limits:

- the
**left-hand**limit (−) is 3.8 - the
**right-hand**limit (+) is 1.3

And the ordinary limit **'does not exist'**

## Are limits only for difficult functions?

Limits can be used even when we **know the value when we get there**! Nobody said they are only for difficult functions.

### Example:

*lim***x→10***x***2** = 5

We know perfectly well that 10/2 = 5, but limits can still be used (if we want!)

## Approaching Infinity

Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.

### Let's start with an interesting example.

Question: What is the value of 1∞ ? |

Answer: We don't know! |

### Why Don't We Know?

The simplest reason is that Infinity is not a number, it is an idea. Gremlins inc. – agents of chaos download for mac os.

So *1***∞** is a bit like saying *1***beauty** or *1***tall**.

Maybe we could say that *1***∞**= 0, .. but that is a problem too, because if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?

In fact *1***∞** is known to be **undefined**.

## 1.2 Limits Algebraicallyap Calculus 2nd Edition

### But We Can Approach It!

So instead of trying to work it out for infinity (because we can't get a sensible answer), let's try larger and larger values of x:

x | 1x |

1 | 1.00000 |

2 | 0.50000 |

4 | 0.25000 |

10 | 0.10000 |

100 | 0.01000 |

1,000 | 0.00100 |

10,000 | 0.00010 |

Now we can see that as x gets larger, ** 1x** tends towards 0

We are now faced with an interesting situation:

- We can't say what happens when x gets to infinity
- But we can see that
is*1***x****going towards 0**

We want to give the answer '0' but can't, so instead mathematicians say exactly what is going on by using the special word 'limit'.

The **limit** of ** 1x** as x approaches Infinity is

**0**

And write it like this:

*lim***x→∞***1***x** = 0

## 1.2 Limits Algebraicallyap Calculus Frq

In other words:

As x approaches infinity, then ** 1x** approaches 0

It is a mathematical way of saying *'we are not talking about when x=∞, but we know as x gets bigger, the answer gets closer and closer to 0'*.

Read more at Limits to Infinity.

## Solving!

## 1.2 Limits Algebraicallyap Calculus 14th Edition

We have been a little lazy so far, and just said that a limit equals some value because it **looked like it was going to**.

## 1.2 Limits Algebraicallyap Calculus Calculator

That is not really good enough! Read more at Evaluating Limits.