Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{x \rightarrow a} \sqrt{f^{2}(x)+g^{2}(x)}$$

Step-by-Step Solution

Answer

The limit is \( \sqrt{10} \).

Step 1: Understanding the expression inside the limit

We start by considering the expression inside the limit: \( \sqrt{f^{2}(x)+g^{2}(x)} \). This involves finding the square root of the sum of the squares of two functions.

Step 2: Finding limits of individual squared functions

Since \( \lim _{x \rightarrow a} f(x) = 3 \), we know \( \lim _{x \rightarrow a} f^{2}(x) = 3^{2} = 9 \). Similarly, since \( \lim _{x \rightarrow a} g(x) = -1 \), it follows that \( \lim _{x \rightarrow a} g^{2}(x) = (-1)^{2} = 1 \).

Step 3: Adding the limits of the squared functions

Add the limits of the squared functions: \( \lim _{x \rightarrow a} f^{2}(x) + \lim _{x \rightarrow a} g^{2}(x) = 9 + 1 = 10 \).

Step 4: Finding the limit of the square root of the sum

Finally, take the square root of the combined limit: \( \lim _{x \rightarrow a} \sqrt{f^{2}(x) + g^{2}(x)} = \sqrt{10} \).

Key Concepts

Continuity

Continuity is an essential property of functions in calculus. A function is continuous at a point if you can draw it without picking up your pencil at that point. Formally, a function \( f(x) \) is continuous at \( x = a \) when three conditions are met:

• \( f(a) \) is defined, meaning the function has a real value at \( a \).
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The value of the function at \( a \), \( f(a) \), is equal to the limit as \( x \) approaches \( a \): \( \lim_{x \to a} f(x) = f(a) \).

Understanding continuity helps in evaluating limits, especially when dealing with complex functions. When each component of a function is continuous at a certain point, the whole function often remains continuous at that point. This concept is vital when transitioning from finding limits to ensuring the continuity of complex expressions, such as those involving squared functions or square roots.

Squared Functions

Squared functions involve the expression \( f(x)^2 \), where each output of the function \( f(x) \) is multiplied by itself. Consider the limit of a squared function: if \( \lim_{x \to a} f(x) = L \), then \( \lim_{x \to a} f(x)^2 = L^2 \).
When working with squared functions, remember that:

• The square of a function is always non-negative, meaning it won't produce negative results, even if \( f(x) \) itself is negative.
• Squaring a negative value results in a positive value, as seen with \( (-1)^2 = 1 \).
• Such transformations often simplify expressions and make calculations easier, as they convert differences into sums.

This concept is particularly useful when dealing with real limits and checking continuity, as squaring eliminates negative signs, thus ensuring a positive contribution from each part of the function. Squared functions play a crucial role in a variety of mathematical concerns, including geometry and statistical calculations.

Square Root

The square root function is one of the most fundamental operations in mathematics. Taking the square root of a number \( y \) means finding a number \( x \) such that \( x^2 = y \). Its importance lies in:

• Only non-negative numbers have real square roots in the domain of real numbers.
• The square root reverses the squaring operation, which can help in solving equations and simplifying expressions.
• In the context of limits, finding the limit of a square root function often involves ensuring the expression inside the root is non-negative. If \( \lim_{x \to a} (f(x)^2 + g(x)^2) \) is always positive, \( \lim_{x \to a} \sqrt{f(x)^2 + g(x)^2} \) exists and equals \( \sqrt{L} \), where \( L \) is the limit of the expression inside.

Understanding square roots is crucial for ensuring accuracy in calculations involving distance, growth, and transformation, particularly in fields like physics and engineering, where these parameters often change continuously.