Question

Evaluate the following limits. $$\lim _{(x, y, z) \rightarrow(1, \ln 2,3)} z e^{x y}$$

Short answer

Question: Find the limit of the function $$ze^{xy}$$ as $$(x, y, z)$$ approaches $$(1, \ln 2, 3)$$. Answer: The limit of the function is 6.

Step-by-step solution

Write down the function and the limit

We have to evaluate the limit of the function $$ze^{xy}$$ as $$(x, y, z)$$ approaches $$(1, \ln 2, 3)$$. So we can write this as: $$\lim_{(x, y, z) \rightarrow (1, \ln 2, 3)} ze^{xy}$$

Substitute the values of the variables in the function

Now substitute the values of x, y, and z with 1, \(\ln 2\), and 3, respectively, into the function: $$3e^{1 \cdot \ln 2}$$

Simplify the expression

Next, simplify the expression, using the property of exponentials that \(a^{\ln_b{a}}=b\). Here, we have \(e^{\ln 2} = 2\), so the expression becomes: $$3 \cdot 2$$

Evaluate the expression

Finally, evaluate the expression to find the value of the limit: $$3 \cdot 2 = 6$$ Thus, the limit of the function $$ze^{xy}$$ as $$(x,y,z)$$ approaches $$(1, \ln 2, 3)$$ is 6.