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Chapter 15: Problem 75

Use the method of your choice to evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,2)} \frac{x^{2}+2 x y-x+y^{2}-y-6}{x+y-3}$$

Short Answer

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Question: Find the limit of the given function as (x, y) approaches (1, 2): $$\lim_{(x, y) \rightarrow (1, 2)} \frac{x^2 + 2xy - x + y^2 - y - 6}{x + y - 3}$$

Step by step solution

01

Direct Substitution

$$\lim_{(x, y) \rightarrow (1, 2)} \frac{x^2 + 2xy - x + y^2 - y - 6}{x + y - 3}$$ Substitute (1, 2) into the function: $$\frac{1^2 + 2(1)(2) - 1 + 2^2 - 2 - 6}{1 + 2 - 3}$$
02

Simplify the expression

$$\frac{1 + 4 - 1 + 4 - 2 - 6}{3 - 3}$$
03

Check if the denominator equals zero and take the limit if possible

Notice that the denominator is equal to 0: $$3 - 3 = 0$$ Since the denominator is 0, we cannot find the limit using direct substitution. We need to try another method.
04

Try factoring numerator and cancel terms

The numerator of the expression can be factored into \(((x + y) -3)(x - 1)\). Now rewrite the whole expression and factor the numerator: $$\lim_{(x, y) \rightarrow (1, 2)} \frac{((x + y) -3)(x - 1)}{x + y - 3}$$ Since (x+y)-3 is common to both the numerator and the denominator, we can cancel it out: $$\lim_{(x, y) \rightarrow (1, 2)} (x - 1)$$
05

Direct substitution on simplified expression

Now we can use direct substitution again: $$\lim_{(x, y) \rightarrow (1, 2)} (x - 1) = (1) - 1$$
06

Final answer

$$\lim_{(x, y) \rightarrow (1, 2)} \frac{x^2 + 2xy - x + y^2 - y - 6}{x + y - 3} = 0$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Direct Substitution
Direct substitution is the most straightforward method of evaluating limits in calculus. It involves simply substituting the values of the variables directly into the expression. In this case, we substitute \((x, y) = (1, 2)\) into the function. However, it's not always successful. In our example, direct substitution resulted in a 0/0 indeterminate form, meaning we can't determine the limit this way.It's a useful tool, but when faced with indeterminate forms, like zero in the denominator, it's time to try another method.
Factoring
Factoring is a powerful tool when dealing with polynomial expressions, especially when direct substitution fails due to indeterminate forms. In this context, it involves rewriting the expression in a way that allows terms to be simplified or canceled. For our exercise, the numerator, \(x^2 + 2xy - x + y^2 - y - 6\),is factored into:- \(((x + y) - 3)(x - 1)\).By factoring the numerator, we identify common terms with the denominator that can be canceled, simplifying the expression and making it possible to evaluate the limit.
Canceling Terms
Once we've factored the expression, the next step is to simplify by canceling common terms between the numerator and the denominator. Here, \((x + y) - 3\) appears in both, allowing us to cancel it out. This leaves the much simpler expression \(x - 1\), which can easily be evaluated using substitution. Canceling terms is essential to moving past indeterminate forms, as it simplifies the expression to something we can directly deal with.
Multivariable Calculus
Multivariable calculus extends the concepts of calculus to functions of multiple variables. It often involves evaluating limits as both variables approach specific values. The process can be complex, because indeterminate forms, like 0/0, often appear. Therefore, techniques like factoring and canceling are crucial. They help us simplify multivariable functions to a single-variable problem, which can then be solved using direct substitution. Understanding these methods is key to navigating the challenges of multivariable calculus.

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Most popular questions from this chapter

Find the values of \(K\) and \(L\) that maximize the following production functions subject to the given constraint, assuming \(K \geq 0\) and \(L \geq 0\) $$P=f(K, L)=10 K^{1 / 3} L^{2 / 3} \text { for } 30 K+60 L=360$$

Travel cost The cost of a trip that is \(L\) miles long, driving a car that gets \(m\) miles per gallon, with gas costs of \(\$ p /\) gal is \(C=L p / m\) dollars. Suppose you plan a trip of \(L=1500 \mathrm{mi}\) in a car that gets \(m=32 \mathrm{mi} / \mathrm{gal},\) with gas costs of \(p=\$ 3.80 / \mathrm{gal}\) a. Explain how the cost function is derived. b. Compute the partial derivatives \(C_{L}, C_{m^{\prime}}\) and \(C_{p^{\prime}}\). Explain the meaning of the signs of the derivatives in the context of this problem. c. Estimate the change in the total cost of the trip if \(L\) changes from \(L=1500\) to \(L=1520, m\) changes from \(m=32\) to \(m=31,\) and \(p\) changes from \(p=\$ 3.80\) to \(p=\$ 3.85\) d. Is the total cost of the trip (with \(L=1500 \mathrm{mi}, m=32 \mathrm{mi} / \mathrm{gal}\). and \(p=\$ 3.80\) ) more sensitive to a \(1 \%\) change in \(L,\) in \(m,\) or in \(p\) (assuming the other two variables are fixed)? Explain.

Line tangent to an intersection curve Consider the paraboloid \(z=x^{2}+3 y^{2}\) and the plane \(z=x+y+4,\) which intersects the paraboloid in a curve \(C\) at (2,1,7) (see figure). Find the equation of the line tangent to \(C\) at the point \((2,1,7) .\) Proceed as follows. a. Find a vector normal to the plane at (2,1,7) b. Find a vector normal to the plane tangent to the paraboloid at (2,1,7) c. Argue that the line tangent to \(C\) at (2,1,7) is orthogonal to both normal vectors found in parts (a) and (b). Use this fact to find a direction vector for the tangent line.

A snapshot (frozen in time) of a set of water waves is described by the function \(z=1+\sin (x-y),\) where \(z\) gives the height of the waves and \((x, y)\) are coordinates in the horizontal plane \(z=0\) a. Use a graphing utility to graph \(z=1+\sin (x-y)\) b. The crests and the troughs of the waves are aligned in the direction in which the height function has zero change. Find the direction in which the crests and troughs are aligned. c. If you were surfing on one of these waves and wanted the steepest descent from the crest to the trough, in which direction would you point your surfboard (given in terms of a unit vector in the \(x y\) -plane)? d. Check that your answers to parts (b) and (c) are consistent with the graph of part (a).

Challenge domains Find the domain of the following functions. Specify the domain mathematically, and then describe it in words or with a sketch. $$f(x, y, z)=\ln \left(z-x^{2}-y^{2}+2 x+3\right)$$
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