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If $f\left(x_{0}, y_{0}\right)=3,$ what can you say about$$(x, y) \rightarrow\left(x_{0}, y_{0}\right)$$if $f$ is continuous at $\left(x_{0}, y_{0}\right) ?$ If $f$ is not continuous at $\left(x_{0}, y_{0}\right) ?$ Give reasons for your answer.

If $f$ is continuous at $\left(x_{0}, y_{0}\right),$ then$\Rightarrow \lim _{(x, y) \rightarrow\left(x_{0}, y_{0}\right)} f(x, y)$ must equal $f\left(x_{0}, y_{0}\right)=3$If $f$ is not continuous at $\left(x_{0}, y_{0}\right),$ the limit could have any value different from $3 .$ Thus, the limitneed not exist.

Calculus 3

Calculus 1 / AB

Chapter 14

Partial Derivatives

Section 2

Limits and Continuity in Higher Dimensions

Applications of the Derivative

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suppose a face of function it is given that a four x note comma by not it's equals 23 The pose if this continents at the point x not common by not the objective is to determine about the limit expression. Lim X Coma via things, too. It's not comma. Why not? A 4 ft come over since FS continues at the point. X not comma by not definition of limit at the point implies that F is defined at x, not comma by not limit. Ex coma by tends to x Not common. Why not 1/4 ex coma? By exist, Andi equals toe 1/4 ex, not common. Why not? Therefore limit X com abidance to X Not common by note. A four X comma by is equals to afford Exco X not comma by note is equals two F or x, not comma by note, which is equals to three on. If the function if is not continues at its not common by not the limit could have any value different from three. Consider the following example A four x camera is equals toe X plus why divided by eggs coma X coma by is equals to zero comma zero on three government X comma y is not equals to zero comma zero If we move along the line y Z equals two eggs, then for old X is not equals to zero. 1/4 Exco my ex is equals two x plus x divided by two which is equals Toe X divided by two which is equals to do They are four. 1/4 ex coma by is equals toe as ex coma by is tends to zero comma zero along bicycles through it eggs but a 40 comma zero is equals to three.

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