Question

Show that f(x, y) = `(x^2 - y^2)/(y - 1)` s continuous at every (x, y) ∈ R² 

Answer 1

Let (a, b) ∈ R² be an arbitrary point.

We shall investigate the continuity of f at (a,b).

That is, we shall check if all the three conditions for continuity hold for f at (a, b)

To check first condition, note that

f(a, b) = `(a^2 + "b"^2)/("b"^2 + 1)` is defined

Next we want to find if `lim_((x,  y) ->("a",  "'b"))` f(x, y) exist or not

So we calculate `lim_((x,  y) -> ("a",  "b"))` x² – y² = a² – b² and `lim_((x,  y) -> ("a",  "b"))` y² + 1 = b² + 1

By the properties of limit we see that

`lim_((x,  y) -> ("a",  "b"))` f(x, y) = `(x^2 - y^2)/("b"^2 + 1)`

=`("a"^2 + "b"^2)/("b"^2 + 1)`

= f(a, b)

= L exists

Now, we note that `lim_((x,  y) -> ("a",  "b"))` f(x, y)

= L

= f(a, b).

Hence f satisfies all the there conditions for continuity of f at (a, b).

Since (a, b) is an arbitrary point in R²

We conclude that f is continuous at every point of R²