Problem 10 - Calculus for Biology and Medicine

Question

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(-1,1)} \frac{x^{2}+y}{2 x+y}\)

Step-by-Step Solution

Answer

The limit is -2.

Step 1: Substitute the values into the function

To find the limit as \((x, y) \rightarrow (-1, 1)\), first substitute \(x = -1\) and \(y = 1\) directly into the expression: \(\frac{x^2 + y}{2x + y}\). This gives \(\frac{(-1)^2 + 1}{2(-1) + 1}\).

Step 2: Evaluate the expression

Calculate the numerator and the denominator after substituting. The numerator becomes \(1 + 1 = 2\) while the denominator becomes \(-2 + 1 = -1\).

Step 3: Simplify the fraction

Now, simplify the fraction \(\frac{2}{-1}\) to get \(-2\).

Step 4: Conclude the limit calculation

Having simplified the expression, the limit as \((x, y) \rightarrow (-1, 1)\) of \(\frac{x^{2}+y}{2x+y}\) is \(-2\).

Key Concepts

Properties of Limits

Understanding the properties of limits is crucial for solving calculus problems efficiently. These properties allow us to handle expressions in manageable pieces and move towards a solution step by step with confidence.

Consider some basic properties of limits:

• **Scalar Multiplication**: The limit of a constant times a function is the constant times the limit of the function.
• **Sum or Difference**: The limit of a sum is the sum of the limits, and similarly, the limit of a difference is the difference of the limits.
• **Product**: The limit of a product is the product of the limits.
• **Quotient**: If the limit of the denominator is not zero, the limit of a quotient is the quotient of the limits.

When you have expressions like \[\lim _{(x, y) \rightarrow(-1,1)} \frac{x^{2}+y}{2 x+y}\], applying these properties allows us to simplify the expression before evaluating it directly.

Substitution Method

The substitution method is a straightforward approach to calculate limits. In this exercise, you're finding the limit of a function as \((x, y) \rightarrow (-1, 1)\). The substitution method involves directly substituting these values into the given function.

By substituting

• \(x = -1\)
• \(y = 1\)

we plug them into the expression \(\frac{x^2 + y}{2x + y}\). Generally, you want to ensure that substituting values doesn't lead to an undefined form, like division by zero. If everything checks out, this method can quickly give you the result.

Numerator and Denominator Evaluation

Once you've substituted values into the function, the next step is to evaluate the numerator and the denominator separately. It makes the process more organized and helps to identify any potential issues with zero divisions or undefined forms.

For example, evaluating the expression \(\frac{(-1)^2 + 1}{2(-1) + 1}\) after substitution:

The **numerator** becomes \(1 + 1 = 2\).

The **denominator** becomes \(-2 + 1 = -1\).

Carefully handling each part ensures that any errors are caught early, which can prevent complications in the fraction simplification step.

Fraction Simplification

After evaluating the numerator and the denominator, the final step is to simplify the fraction. Simplification involves dividing the numerator by the denominator to reach the simplest form of the expression.

In our example, you have:\[\frac{2}{-1} = -2\]
Here, division of the two numbers simplifies the given fraction significantly and demonstrates that the limit of the function as \((x, y) \rightarrow (-1, 1)\) is \(-2\). By ensuring each part is calculated and simplified correctly, the final outcome provides the calculated limit cleanly and accurately.