Math / Calculus

QuestionSuppose that $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=7$ . What can you say about the value of $f(5,2)$ ? We can say $f(5,2)=\infty$ . We can say $f(5,2)$ is an open point. We can say $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=f(5,2)=7$ . We can say $\underset{(x, y) \rightarrow(5,2)}{ } f(x, y)=f(5,2)=\infty$ . In general, nothing can be said about the value of $f(5,2)$ .
What if $f$ is continuous? We can say $f(5,2)=\infty$ . We can say $f(5,2)$ is an open point. We can say $(x, y) \rightarrow(5,2)$ . We can say $\lim _{(x, y) \rightarrow(5,2)} f(x, y)=f(5,2)=\infty$ . In general, nothing can be said about the value of $f(5,2)$ .

Studdy Solution

STEP 1

What is this asking? If we know a function approaches a value as its input approaches a point, can we say anything about the function's actual value at that point?
What if the function is continuous? Watch out! Don't assume the limit of a function at a point is the same as the function's value at that point!
That's only true if the function is continuous at that point.

STEP 2

1. Limit Definition
2. Continuity Definition
3. Analyze the Problem

STEP 3

Let's remember what a limit means!
The expression $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = 7$ means as $(x, y)$ gets really close to $(5, 2)$ , $f(x, y)$ gets really close to 7.
Notice, we're talking about what happens near $(5, 2)$ , not at $(5, 2)$ .

STEP 4

Now, a function is continuous at a point if the limit of the function as the input approaches that point is equal to the function's value at that point.
In math terms, $f(x, y)$ is continuous at $(5, 2)$ if $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = f(5, 2)$ .

STEP 5

The problem tells us $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = 7$ .
This tells us nothing about $f(5, 2)$ !
The function could be defined at $(5, 2)$ , undefined at $(5, 2)$ , or even be a completely different value like $f(5, 2) = 1000$ !

STEP 6

However, if $f$ is continuous, then by definition, the limit and the function's value at that point must be the same!
So, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ !

STEP 7

If we only know the limit of $f(x, y)$ as $(x, y)$ approaches $(5, 2)$ is 7, we can't say anything about $f(5, 2)$ .
But, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ .

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Studdy Solution

STEP 1

STEP 2

1. Limit Definition
2. Continuity Definition
3. Analyze the Problem

STEP 3

Let's remember what a limit means!
The expression $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = 7$ means as $(x, y)$ gets really close to $(5, 2)$ , $f(x, y)$ gets really close to 7.
Notice, we're talking about what happens near $(5, 2)$ , not at $(5, 2)$ .

STEP 4

Now, a function is continuous at a point if the limit of the function as the input approaches that point is equal to the function's value at that point.
In math terms, $f(x, y)$ is continuous at $(5, 2)$ if $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = f(5, 2)$ .

STEP 5

The problem tells us $\lim_{(x, y) \rightarrow (5,2)} f(x, y) = 7$ .
This tells us nothing about $f(5, 2)$ !
The function could be defined at $(5, 2)$ , undefined at $(5, 2)$ , or even be a completely different value like $f(5, 2) = 1000$ !

STEP 6

However, if $f$ is continuous, then by definition, the limit and the function's value at that point must be the same!
So, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ !

STEP 7

If we only know the limit of $f(x, y)$ as $(x, y)$ approaches $(5, 2)$ is 7, we can't say anything about $f(5, 2)$ .
But, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ .

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Studdy Solution

STEP 1

STEP 2

1. Limit Definition2. Continuity Definition3. Analyze the Problem

STEP 3

STEP 4

STEP 5

STEP 6

However, if fff is **continuous**, then by definition, the limit and the function's value at that point *must* be the same!So, if fff is continuous at (5,2)(5, 2)(5,2), then we *know* f(5,2)=7f(5, 2) = 7f(5,2)=7!

STEP 7

If we only know the limit of f(x,y)f(x, y)f(x,y) as (x,y)(x, y)(x,y) approaches (5,2)(5, 2)(5,2) is **7**, we can't say *anything* about f(5,2)f(5, 2)f(5,2).But, if fff is **continuous** at (5,2)(5, 2)(5,2), then we know f(5,2)=7f(5, 2) = 7f(5,2)=7.

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1. Limit Definition
2. Continuity Definition
3. Analyze the Problem

However, if $f$ is continuous, then by definition, the limit and the function's value at that point must be the same!
So, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ !

If we only know the limit of $f(x, y)$ as $(x, y)$ approaches $(5, 2)$ is 7, we can't say anything about $f(5, 2)$ .
But, if $f$ is continuous at $(5, 2)$ , then we know $f(5, 2) = 7$ .