Calculating a multivariable limit $\lim_{(x,y,z)->(1,\ln3,\pi)}e^{xy\cos(z)}$

Question

$$\lim_{(x,y,z)->(1,\ln3,\pi)}e^{xy\cos(z)}$$

My thoughts are just plugging in the values of $x,y,z$ in $e^{xy\cos(z)}$ and then evaluate the limit.

Can anyone please explains this.

Answer 1

You are correct. Since, when evaluated we find it equal to $e^{\ln3\cos(\pi)}=\frac{1}{3}$, it follows that this is the limit of the function.

When we are trying to solve a multivariable limit, the first thing we should usually do is evaluate it at the point in question. If it is undefined, then we usually must make use of some clever inequalities or other technique. However if the function is defined at that point, then whatever it is defined as is the limit at that point.

Answer 2

Function $f=e^{xy\cos(z)}$ is continuous.

By definition of continuity $f(a)=\lim_{x\rightarrow a} f$

So you can simply put the values in the expression.