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Limits
In order to really understand Calculus, we first must understand a limit of a function at a point. The notion of a limit is the value the function approaches (y) as we approach a particular value of x. In order for a function to have a limit at a point the value of the function must only **approach the same value from the left and the right**, not necessarily the value at the point.
More formally:

**lim x->A+ f(x) = lim x->A- f(x)**

We will look at limits from a graph, a table and algebraically with a piecewise function.

Limits from a Graph
**Example 1**

**Limit at A**

Since, lim x -> A+ = 2 and lim x -> A- = 2. OR more simply lim x -> A+ = lim x -> A- , then lim x -> A = 2. The fact that the value of the function at A = 5 has no bearing upon the limit. The limit is what the function APPROACHES, not what the function value is at x = A.

**Limit at B**

Since, lim x -> B+ = 2 and lim x -> B- = 2, since lim x -> B+ = lim x -> B- , then lim x -> B = 2.

**Example 2**

**Limit at A**

Since, lim x -> A+ = 3 and lim x -> A- = 1. Since, lim x -> A+ = 3 â‰ lim x -> A- , then no limit exists a x = A.

**Limits at Q and R**

Point Q only has a limit from the right which is 1, no limit from the left exists since the function does not exist to the left of Q. In other words, point Q only has a right-handed limit which is 1.

Similarly, point R only has a limit from the left which is 3, no limit from the right exists since the function does not exist to the right of R. In other words, point R only has a left-handed limit which is 3.

**Example 3**

**Limit at 3**

Since, lim x -> 3+ = 0 and lim x -> 3- = 0. Since, lim x -> 3+ = lim x -> 3- , then limit x -> 3 = 0

Limits from a Table
Limits Algebraically for a Piecewise Function I
Limits Algebraically for a Piecewise Function II
More Here

**approach the same value from the left and the right**, not necessarily the value at the point.

More formally:

**lim x->A+ f(x) = lim x->A- f(x)**

We will look at limits from a graph, a table and algebraically with a piecewise function.

**Example 1**

**Limit at A**

Since, lim x -> A+ = 2 and lim x -> A- = 2. OR more simply lim x -> A+ = lim x -> A- , then lim x -> A = 2. The fact that the value of the function at A = 5 has no bearing upon the limit. The limit is what the function APPROACHES, not what the function value is at x = A.

**Limit at B**

Since, lim x -> B+ = 2 and lim x -> B- = 2, since lim x -> B+ = lim x -> B- , then lim x -> B = 2.

**Example 2**

**Limit at A**

Since, lim x -> A+ = 3 and lim x -> A- = 1. Since, lim x -> A+ = 3 â‰ lim x -> A- , then no limit exists a x = A.

**Limits at Q and R**

Point Q only has a limit from the right which is 1, no limit from the left exists since the function does not exist to the left of Q. In other words, point Q only has a right-handed limit which is 1.

Similarly, point R only has a limit from the left which is 3, no limit from the right exists since the function does not exist to the right of R. In other words, point R only has a left-handed limit which is 3.

**Example 3**

**Limit at 3**

Since, lim x -> 3+ = 0 and lim x -> 3- = 0. Since, lim x -> 3+ = lim x -> 3- , then limit x -> 3 = 0

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Continuity
The next important fundamental concept is continuity or determining if a function is continuous.

In order for a function to be continuous at a particular point, it must first have a limit at the point – if it does *not* have a limit, it **can not be** continuous. Secondly, the value of the function must also equal the limit. More formally,** lim x->A+ f(x) = lim x->A- f(x) and = f(A).** In other words, there cannot be any breaks, jumps or gaps in the function or graph.
More informally, a simple way to conceptually understand the notion of continuity is that when drawing or tracing the graph, you never lift your pencil when a function is continuous. If you lift your pencil, it is not a continuous function. Please note that a function can be continuous for all values on an interval, except for a single point.

Special note: The “lifting your pencil test” is not an acceptable means of proving continuity, just an aide for conceptually understanding continuity. For showing continuity at a point B on the function f(x), you must show that lim x->B+ f(x) = lim x->B- f(x)** and **= f(B) or you can simply say there is a break in the graph or in another case an asymptote exists exists and it is not continuous.

Determining Continuity from a Graph
Determining Continuity from a Table
Determining Continuity Algebraically for a Piecewise Function I
More Here

In order for a function to be continuous at a particular point, it must first have a limit at the point – if it does

*not*have a limit, it

**can not be**continuous. Secondly, the value of the function must also equal the limit. More formally,

**lim x->A+ f(x) = lim x->A- f(x) and = f(A).**In other words, there cannot be any breaks, jumps or gaps in the function or graph.

More informally, a simple way to conceptually understand the notion of continuity is that when drawing or tracing the graph, you never lift your pencil when a function is continuous. If you lift your pencil, it is not a continuous function. Please note that a function can be continuous for all values on an interval, except for a single point.

Special note: The “lifting your pencil test” is not an acceptable means of proving continuity, just an aide for conceptually understanding continuity. For showing continuity at a point B on the function f(x), you must show that lim x->B+ f(x) = lim x->B- f(x)** and **= f(B) or you can simply say there is a break in the graph or in another case an asymptote exists exists and it is not continuous.

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Differentiabilty
The next important fundamental concept is differentiabilty or determining if a function is differentiable. In order for a function to be differentiable it must first be continuous (hence, must also have a limit) – if it is not continuous, it can’t be differentiable. Secondly, the SLOPE of the function must be the same from both sides at a particular point.
More informally, a simple way to conceptually understand the notion of differentiability is only one tangent line can be drawn to the graph at any point or a graph must be smooth and can not have and sharp points, such as the absolute value function.

Please note that a function can be differentiable for all values on an interval, except for a single point.

Special note: To prove that a function is not differentiable, you must show that it is either not continuous or that lim x-> A+ **f ‘(x) **= lim x-> A- **f ‘(x).**

1. If a vertical asymptote exists at a point, it is NOT continuous and therefore not differentiable either.

2. The absolute value function is continuous at all points and differentiable at all points except at the bottom (or top if it is negative) of the v-shaped graph. Why? The **slopes** are different as you approach the “sharp v” from both sides.

Determining Differentiabilty from a Graph
Determining Differentiabilty Algebraically for a Piecewise Function I
More Here

More informally, a simple way to conceptually understand the notion of differentiability is only one tangent line can be drawn to the graph at any point or a graph must be smooth and can not have and sharp points, such as the absolute value function.

Please note that a function can be differentiable for all values on an interval, except for a single point.

Special note: To prove that a function is not differentiable, you must show that it is either not continuous or that lim x-> A+ **f ‘(x) **= lim x-> A- **f ‘(x).**

1. If a vertical asymptote exists at a point, it is NOT continuous and therefore not differentiable either.

2. The absolute value function is continuous at all points and differentiable at all points except at the bottom (or top if it is negative) of the v-shaped graph. Why? The **slopes** are different as you approach the “sharp v” from both sides.