Problem 11
Question
Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(0,1)} \frac{2 x y-3}{x^{2}+y^{2}+1}\)
Step-by-Step Solution
Verified Answer
The limit as \( (x, y) \rightarrow (0,1) \) is \(\frac{-3}{2}\).
1Step 1: Substitute the limit values into the expression
First, test the expression by substituting the limit values directly into the function. Substitute \(x = 0\) and \(y = 1\) into the expression \(\frac{2xy - 3}{x^2 + y^2 + 1}\). This gives \(\frac{2(0)(1) - 3}{0^2 + 1^2 + 1} = \frac{-3}{2}\).
2Step 2: Simplify the expression after substitution
Evaluate what happens when the limit values are substituted into the expression: \(\frac{-3}{2}\). The function doesn't become undefined or show any indeterminate form like \(\frac{0}{0}\).
3Step 3: Identify the limit
Since the substitution didn't result in any undefined form, and the expression simplifies directly to a fraction, the limit exists and is equal to \(\frac{-3}{2}\).
Key Concepts
Limit SubstitutionProperties of LimitsEvaluating Limits
Limit Substitution
Limit substitution is an essential tool for evaluating limits, especially when the function involved is continuous around the point we are considering. The idea is simple: you replace the variables in the expression with their limit values.
This straightforward method helps to see if the direct substitution yields a valid result. If the result is a clear number and doesn't lead to anything undefined, like division by zero, you have your answer. In our exercise, substituting the given values of \(x = 0\) and \(y = 1\) directly into the expression \(\frac{2xy - 3}{x^2 + y^2 + 1}\) simplifies to \(\frac{-3}{2}\).
By performing this substitution, you found a clear answer without needing further manipulation. This is because the expression doesn't present any discontinuities or indeterminate forms at the substitution values.
This straightforward method helps to see if the direct substitution yields a valid result. If the result is a clear number and doesn't lead to anything undefined, like division by zero, you have your answer. In our exercise, substituting the given values of \(x = 0\) and \(y = 1\) directly into the expression \(\frac{2xy - 3}{x^2 + y^2 + 1}\) simplifies to \(\frac{-3}{2}\).
By performing this substitution, you found a clear answer without needing further manipulation. This is because the expression doesn't present any discontinuities or indeterminate forms at the substitution values.
Properties of Limits
Properties of limits are fundamental rules that facilitate the computation of limits in calculus. These properties can simplify your work significantly when dealing with more complex expressions. Some key properties include:
In applying these properties, it's crucial to ensure that the limits involved are defined. If substitution yields valid numerical values, as it did in the exercise, these properties confirm the correctness of the result. For our example, the result of substitution directly aligned with the properties since the function is continuous and the denominator is non-zero after substitution.
- The Sum Rule: \(\lim_{x \to a}(f(x) + g(x)) = \lim_{x \to a}f(x) + \lim_{x \to a}g(x)\)
- The Product Rule: \(\lim_{x \to a}(f(x)g(x)) = (\lim_{x \to a}f(x))(\lim_{x \to a}g(x))\)
- The Quotient Rule: \(\lim_{x \to a}\frac{f(x)}{g(x)} = \frac{\lim_{x \to a}f(x)}{\lim_{x \to a}g(x)}\), provided \(\lim_{x \to a}g(x) eq 0\)
In applying these properties, it's crucial to ensure that the limits involved are defined. If substitution yields valid numerical values, as it did in the exercise, these properties confirm the correctness of the result. For our example, the result of substitution directly aligned with the properties since the function is continuous and the denominator is non-zero after substitution.
Evaluating Limits
Evaluating limits is all about determining the behavior of a function as the input approaches a particular point from both sides. This process can involve different techniques depending on the complexity of the expression.
In some cases, like in our example, direct substitution is enough. This means the expression immediately simplifies to a finite number without further simplification. However, when direct substitution results in forms like \(\frac{0}{0}\) or \(\infty - \infty\), more advanced techniques such as factoring, rationalizing, or using the L'Hôpital's Rule might be necessary.
It's important to verify whether the resulting value makes sense in the context of the problem. By carefully following practical methods like substitution and applying limit laws, you can effectively evaluate the limit of a function like we did to find that \(\lim_{(x, y) \rightarrow(0,1)} \frac{2xy - 3}{x^2 + y^2 + 1}\) results in \(\frac{-3}{2}\).
In some cases, like in our example, direct substitution is enough. This means the expression immediately simplifies to a finite number without further simplification. However, when direct substitution results in forms like \(\frac{0}{0}\) or \(\infty - \infty\), more advanced techniques such as factoring, rationalizing, or using the L'Hôpital's Rule might be necessary.
It's important to verify whether the resulting value makes sense in the context of the problem. By carefully following practical methods like substitution and applying limit laws, you can effectively evaluate the limit of a function like we did to find that \(\lim_{(x, y) \rightarrow(0,1)} \frac{2xy - 3}{x^2 + y^2 + 1}\) results in \(\frac{-3}{2}\).
Other exercises in this chapter
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When the initial parasitoid density is \(P_{0}=0\), the negative binomial model reduces to $$ N_{t+1}=b N_{t} $$ as shown in the previous problem. For which val
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