• Delta epsilon definition: means thatThis means that can be arbitrarily close to by making sufficiently close (but not equal) to

Evaluating limits

• Discontinuity: try both sides and see if limit matches
• Try cancelling out the denominator by factoring or expanding
• Plug in directly
• Squeeze theorem if you know the function if bounded

left = -2; right = 5;
bottom = -2; top = 5;
---
f(x) = (x - 3)^3 + (x - 3)^2 + 2
y = 2 \{0 < x <= 3\} | dashed | #388c46
x = 3 \{0 < y <= 2\} | dashed | #388c46
(3, 0) | label: a | #388c46
x = 4 \{0 < y <= 4\} | dashed | #c74440
(4, 4) | label: (x, f(x)) | #c74440
(4, 0) | label: x | #c74440

As goes to , approaches , so we say limit of

left = -2; right = 5;
bottom = -2; top = 5;
---
f(x) = (x - 3)^3 + (x - 3)^2 + 1 \{x <= 3\} | #2d70b3
(3, 1) | open | #2d70b3
(3, 2) | point | #2d70b3
f(x) = (x - 3)^3 + (x - 3)^2 + 2 \{x > 3\} | #2d70b3
y = 1 \{0 < x <= 3\} | dashed | #388c46
y = 2 \{0 < x <= 3\} | dashed | #388c46
x = 3 \{0 < y <= 2\} | dashed | #388c46
(3, 0) | label: a | #388c46

• If approaches from the left to , then
• If approaches from the right to , then

The statement means that for all positive numbers , there exists a number such that:

(See implication)

This means that can be arbitrarily close to by making sufficiently close (but not equal) to

does not depend on the value at . For example

left = -2; right = 5;
bottom = -5; top = 5;
---
y = (x - 2)^2 - 2
(2, -2) | open
(2, 2) | point | label: a

, but

We must show that for any , there exists a such that

Working backwards again,

Assuming the hypothesis, then

is continuous at if and only if

(See logical and)

Where is the right side, and approaches from the left. For piecewise functions, we need to check of the two limits match.

The function is continuous at

It's your lucky day, you can just plug in directly
- Prof. JINTAO Deng

There is a discontinuity in the denominator. If we approach the limit from the right, we get , and if we approach from the left, we get . Therefore, the limit does not exist at .

We can factor the numerator

It's not your lucky day, but you can KILL the troublemaker
- Prof. JINTAO Deng

This requires some algebra and expanding using pascals triangle
For ,

It follows that

Behaves strangely near .

Trouble maker . You can't do anything, it is not your lucky day.
- Prof. JINTAO Deng

However, it is bounded:

So we can state that

So from the squeeze theorem, we can conclude that:

We can split into two cases

Since the limits don't match, the limit does not exist