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Chapter 13: Problem 79

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0,2)}(2 x y)^{x y}$$

Short Answer

Expert verified
Question: Determine the value of the limit as (x, y) approaches (0, 2) for the function \((2xy)^{xy}\). Answer: The limit of the given function as (x, y) approaches (0, 2) is 1.

Step by step solution

01

Write the given function in the exponential form

We have the function \((2xy)^{xy}\), which can be rewritten as \(e^{xy \ln(2xy)}\) using the property \(a^b = e^{b\ln(a)}\).
02

Apply substitution

Let \(t = xy\). Then, when \((x, y) \rightarrow (0, 2)\), \(t \rightarrow 0\). So, now consider the function \(e^{t\ln(2t)}\), and we will now find the limit as \(t \rightarrow 0\) instead.
03

Rewrite the limit

We have now the function \(e^{t\ln(2t)}\). We need to find \(\lim_{t \rightarrow 0} e^{t\ln(2t)}\). Using properties of limits, we can rewrite the function as \(e^{\lim_{t \rightarrow 0} (t\ln(2t))}\).
04

Simplify the limit

Now we must find the limit of the exponent \(\lim_{t \rightarrow 0} (t\ln(2t))\). This is an indeterminate form of type \(0\cdot\infty\). To simplify, we apply L'H么pital's rule to the equivalent expression \(\frac{\ln(2t)}{\frac{1}{t}}\).
05

Apply L'H么pital's Rule

Using L'H么pital's Rule, we differentiate both the numerator and denominator. \(\frac{d(\ln(2t))}{dt} = \frac{1}{t}\) and \(\frac{d(\frac{1}{t})}{dt} = -\frac{1}{t^2}\). The limit becomes \(\lim_{t \rightarrow 0} \left(\frac{\frac{1}{t}}{-\frac{1}{t^2}}\right)\), which can be rewritten as \(\lim_{t \rightarrow 0} (-t)\).
06

Evaluate the limit

The limit \(\lim_{t \rightarrow 0} (-t) = 0\).
07

Find the final answer

We now know that the limit in the exponent is \(0\). The final answer is thus \(e^{\lim_{t \rightarrow 0} (t\ln(2t))} = e^0 = 1\). Therefore, the limit of the given function is: $$\lim_{(x, y) \rightarrow (0, 2)} (2xy)^{xy} = 1$$.

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Most popular questions from this chapter

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1.\) $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$

Find the domains of the following functions. Specify the domain mathematically and then describe it in words or with a sketch. $$g(x, y, z)=\frac{10}{x^{2}-(y+z) x+y z}.$$

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