Problem 1 - Calculus for Biology and Medicine

Question

Use the properties of limits to calculate the following limits: \(\lim _{(x, y) \rightarrow(1.0)}\left(x^{2}-3 y^{2}\right)\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the limit components

We need to calculate the limit of the expression \(x^2 - 3y^2\) as \((x, y)\) approaches \((1, 0)\). Here, identify the components as \(x^2\) and \(-3y^2\) which we will evaluate separately.

Step 2: Calculate the limits separately

First, calculate \(\lim_{(x, y) \to (1, 0)} x^2 = 1^2 = 1\). Then, calculate \(\lim_{(x, y) \to (1, 0)} -3y^2 = -3(0)^2 = 0\).

Step 3: Apply the limit properties

Using the linearity of limits, combine the results from Step 2: \(\lim_{(x, y) \to (1, 0)} (x^2 - 3y^2) = \lim_{(x, y) \to (1, 0)} x^2 - \lim_{(x, y) \to (1, 0)} 3y^2 = 1 - 0 = 1\).

Step 4: State the final limit conclusion

Thus, the limit \(\lim_{(x, y) \to (1, 0)} (x^2 - 3y^2)\) is equal to 1.

Key Concepts

Multivariable Calculus

Multivariable Calculus deals with functions of more than one variable, such as functions with input values that are pairs, triples, or even more numbers. These variables can be seen as coordinates in space. The main principle involves exploring how changes in these input variables affect the output. When considering limits in this context, it involves understanding how a multivariable function behaves as its inputs approach a certain point. In our exercise, we examine the function \( f(x, y) = x^2 - 3y^2 \) as the pair \((x, y)\) gets close to the point \((1, 0)\). This involves evaluating how the function values approach a particular outcome, offering insights into the behavior of surfaces and curves in higher dimensions.

Limit Properties

Limit properties are essential in analyzing the behavior of functions as variables approach a specific point. These properties make evaluating complicated expressions more manageable and less error-prone. Here are some useful properties:

• Linearity: This allows us to evaluate limits of sums and differences separately, as long as these individual limits exist. In simple terms, \( \lim_{(x, y) \to (a, b)} [f(x, y) + g(x, y)] = \lim_{(x, y) \to (a, b)} f(x, y) + \lim_{(x, y) \to (a, b)} g(x, y) \).
• Scalar Multiplication: For any constant \(c\), \( \lim_{(x, y) \to (a, b)} c \cdot f(x, y) = c \cdot \lim_{(x, y) \to (a, b)} f(x, y) \).
• Continuity: If the function is continuous at the given point \((a, b)\), then \( \lim_{(x, y) \to (a, b)} f(x, y) = f(a, b) \).

In our exercise, these properties simplify the approach since we can evaluate \(x^2\) and \(-3y^2\) independently and then combine their results, thanks to linearity.

Limit Evaluation Techniques

Evaluating limits in multivariable calculus can sometimes be straightforward, but at other times, it requires specific techniques to process more involved expressions. Here are common techniques for evaluating limits:

• Direct Substitution: When the function is continuous and no undefined form occurs, simply replace each variable with its limit value.
• Path Testing: Sometimes, analyzing limits along different paths (straight paths, curves) helps establish if the limit exists. Different results for different paths indicate the limit does not exist.
• Squeeze Theorem: Useful when bounding the function by two simpler functions with known limits, giving indirect insight into the desired limit.

In the example exercise, direct substitution is effectively used. Since there are no discontinuities, we evaluate \(x^2\) and \(-3y^2\) separately at \((1, 0)\), confirming their limit values by straightforward calculation.