Limit of a Function of Two Variables – Example 1

We are tasked with evaluating the following limit involving two variables:

$$ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} $$

As \((x, y)\) approaches \((0, 0)\), the expression takes on the indeterminate form \( \frac{0}{0} \):

$$ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} = \frac{0}{0} $$

To resolve this, we introduce a change of variables.

Let us define a new variable \( t \) as:

$$ t = x^2 + y^2 $$

This substitution reduces the original two-variable limit to a well-known single-variable limit:

$$ \lim_{t \rightarrow 0} \frac{\sin(t)}{t} $$

It is a standard result that:

$$ \lim_{t \rightarrow 0} \frac{\sin(t)}{t} = 1 $$

Consequently, the limit of the original function \( f(x, y) \) is also equal to 1.

By replacing \( t \) with \( x^2 + y^2 \), we obtain the final result:

$$ \lim_{(x,y) \rightarrow (0,0)} \frac{\sin(x^2+y^2)}{x^2+y^2} = 1 $$

The limit has thus been successfully evaluated.

And so on.