Problem 18
Question
Evaluate the following limits. $$\lim _{(x, y) \rightarrow(1,-1)} \frac{10 x y-2 y^{2}}{x^{2}+y^{2}}$$
Step-by-Step Solution
Verified Answer
Answer: -6
1Step 1: Substitute the limit values
To find the limit as (x, y) approaches (1, -1), we can substitute these values directly into the expression, as the denominator won't be 0:
$$\lim_{(x, y) \rightarrow (1, -1)} \frac{10xy - 2y^2}{x^2 + y^2} = \frac{10(1)(-1) - 2(-1)^2}{(1)^2 + (-1)^2}$$
2Step 2: Simplify the expression
Simplify the expression by carrying out the calculations:
$$= \frac{-10 - 2}{1 + 1}$$
3Step 3: Complete the calculations
Continuing to simplify:
$$= \frac{-12}{2}$$
4Step 4: Calculate the limit
Finally, finish the calculations to find the limit:
$$= -6$$
So the limit as (x, y) approaches (1, -1) is -6.
Key Concepts
Understanding LimitsUnpacking Two-Variable FunctionsExploring Limit Evaluation Techniques
Understanding Limits
In calculus, limits are fundamental to understanding how functions behave as their variables approach certain values. Simply put, a limit illustrates what value a function "approaches" as the inputs get closer to some point.
For a basic function of one variable, you might want to find the limit of a function as \( x \) approaches a value \( a \). In this exercise, however, we're dealing with a two-variable function, where we observe what happens as both \( x \) and \( y \) head toward specific values, such as \( (1, -1) \).
Limits can show us where a function might be undefined or provide insights into the function's behavior at parameter boundaries. In this example, the two-variable function shows how closely it edges toward a particular value when \( x \) and \( y \) converge on \( (1, -1) \).
For a basic function of one variable, you might want to find the limit of a function as \( x \) approaches a value \( a \). In this exercise, however, we're dealing with a two-variable function, where we observe what happens as both \( x \) and \( y \) head toward specific values, such as \( (1, -1) \).
Limits can show us where a function might be undefined or provide insights into the function's behavior at parameter boundaries. In this example, the two-variable function shows how closely it edges toward a particular value when \( x \) and \( y \) converge on \( (1, -1) \).
Unpacking Two-Variable Functions
Two-variable functions add layers of complexity to mathematics. These functions have inputs that involve two variables, typically noted as \( x \) and \( y \). Imagine a landscape with peaks and valleys; two-variable functions describe the surface of this landscape using mathematical language. The function's value depends on where you "stand" on the \( x,y \) plane.
Two-variable limits, like the one in this exercise, assist us in envisioning what the surface looks like at critical points. This gives essential insights into both mathematical and real-world phenomena. We assessed the limit of \( \frac{10xy - 2y^2}{x^2 + y^2} \) as \( (x,y) \) approached \( (1, -1) \), smoothly revealing the surface's tendencies at that point. Understanding the limit will help us recognize symmetry or systematic trends in the function's behavior.
Two-variable limits, like the one in this exercise, assist us in envisioning what the surface looks like at critical points. This gives essential insights into both mathematical and real-world phenomena. We assessed the limit of \( \frac{10xy - 2y^2}{x^2 + y^2} \) as \( (x,y) \) approached \( (1, -1) \), smoothly revealing the surface's tendencies at that point. Understanding the limit will help us recognize symmetry or systematic trends in the function's behavior.
Exploring Limit Evaluation Techniques
Evaluating limits, particularly for two-variable functions, sometimes involves different techniques when basic substitution doesn't work due to a zero in the denominator or if the function lacks continuity.
When directly substituting \( (x,y) = (1,-1) \) into our function produced a valid, non-zero result, this direct substitution became an apt technique. Often, mathematicians employ other methods, like:
When directly substituting \( (x,y) = (1,-1) \) into our function produced a valid, non-zero result, this direct substitution became an apt technique. Often, mathematicians employ other methods, like:
- Approaching the point along different paths: This involves substituting different equations in for \( y \), observing whether limits match from each path.
- Employing polar coordinates in complex cases to simplify function behavior.
- Squeezing theorem for bounding strategies.
Other exercises in this chapter
Problem 17
Find the first partial derivatives of the following functions. $$f(w, z)=\frac{w}{w^{2}+z^{2}}$$
View solution Problem 18
Find all critical points of the following functions. $$f(x, y)=e^{x^{2} y^{2}-2 x y^{2}+y^{2}}$$
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Find an equation of the plane tangent to the following surfaces at the given points. $$z=2+2 x^{2}+\frac{y^{2}}{2} ;\left(-\frac{1}{2}, 1,3\right) \text { and }
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Compute the directional derivative of the following functions at the given point P in the direction of the given vector. Be sure to use a unit vector for the di
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