Question

In Exercises \(47-52,\) discuss the continuity of the function. \(f(x, y, z)=\frac{\sin z}{e^{x}+e^{y}}\)

Short answer

The function \(f(x, y, z)=\frac{\sin z}{e^{x}+e^{y}}\) is continuous for all real values.

Step-by-step solution

Consider the function

We are given the function \(f(x, y, z)=\frac{\sin z}{e^{x}+e^{y}}\). This function is made up of two parts, with the numerator being the sine function of \(z\), which is continuous for all real values. The denominator is a sum of two exponential functions, \(e^{x}\) and \(e^{y}\), which are also continuous for all real values.

Check the denominator

We notice that the denominator could be zero when both \(x\) and \(y\) tend to negative infinity, which is a condition where the function will be undefined. But, as we are working in the real number system, negative infinity is not a number to be concerned about as the function should be defined for all real values of \(x\), \(y\), and \(z\). Therefore, this function is defined and continuous for all real numbers.

Conclusion

Since the numerator and the denominator are both continuous for all real values (not considering the condition when both \(x\) and \(y\) tend to negative infinity), and knowing that the composition of continuous functions is continuous, we can conclude that the function \(f(x, y, z)=\frac{\sin z}{e^{x}+e^{y}}\) itself is continuous for all real values.