Question

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

Short answer

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

Step-by-step solution

Identify the problem

We need to find the limit of the function \( h(x) = \sqrt{f^2(x) + g^2(x)} \) as \( x \to a \). Given the limits of \( f(x) \) and \( g(x) \) as \( x \to a \).

Apply the given limits

From the problem, we know \( \lim_{x \to a} f(x) = 3 \) and \( \lim_{x \to a} g(x) = -1 \). Thus, as \( x \to a \), \( f(x) \) approaches 3 and \( g(x) \) approaches -1.

Calculate the expression inside the square root

Substitute the limits into the expression inside the square root: \[\lim_{x \to a} \left( f^2(x) + g^2(x) \right) = \lim_{x \to a} \left( 3^2 + (-1)^2 \right) = 9 + 1 = 10.\]

Evaluate the overall limit

Take the limit of the square root:\[\lim_{x \to a} \sqrt{f^2(x) + g^2(x)} = \sqrt{\lim_{x \to a} (f^2(x) + g^2(x))} = \sqrt{10}.\]

Final Step: Conclusion

The limit of \( \sqrt{f^2(x) + g^2(x)} \) as \( x \to a \) is \( \sqrt{10} \).

Key concepts

Limit of a Function

To grasp calculus, understanding the concept of the limit of a function is essential. In simple terms, a limit describes how a function behaves as the input approaches a certain point. Imagine driving towards a destination; the limit is the value your speedometer would point to as you reach your target. Calculus uses limits to understand what happens to functions near specific points rather than just at the points themselves.
For the exercise given, we know the limits of two functions: \( \lim _{x \to a} f(x) = 3 \) and \( \lim _{x \to a} g(x) = -1 \). This information means that as \( x \) gets very close to \( a \), \( f(x) \) and \( g(x) \) approach values of 3 and -1, respectively.
The concept of limits allows mathematicians and students to understand and calculate the behavior of functions at points where they may not be easily defined directly. It’s like peeking into the future behavior of the function as \( x \) gets infinitely close to \( a \) without necessarily arriving there.

Square Root of Functions

The square root of functions takes the familiar operation of finding square roots to the level of variable expressions in calculus. Calculating the square root in mathematical functions involves finding which number multiplied by itself gives the original number. When functions are involved, the concept remains the same, but with algebraic expressions under the radical sign.
In the problem we're discussing, we deal with the function \( h(x) = \sqrt{f^2(x) + g^2(x)} \). This means we want to take the square root of the expression formed by squaring \( f(x) \) and \( g(x) \) and then adding them together. This highlights a critical aspect of calculus: much of the work involves manipulating complex expressions to retain their simplicity or see them in a more straightforward way.
Square roots naturally help in understanding situations involving physical distance or any scenarios where the Pythagorean theorem comes into play. Simplifying expressions under a root by using given limits, like in this task, can often make problems much easier to tackle.

Limit Properties

Calculus uses a set of rules for limits that simplify solving problems like our current task. Using these properties, you can find the limits for more complex expressions calculated by combining basic functions. These properties include:

• Sum/Difference Rule: The limit of a sum or difference is the sum or difference of the limits.
• Product Rule: The limit of a product is the product of the limits.
• Quotient Rule: The limit of a quotient (division) is the quotient of the limits, provided the limit of the denominator is not zero.
• Root Rule: The limit of the square root of a function is the square root of the limit of the function.

These properties simplify complex calculations and are all applied without directly affecting the end result’s accuracy. In our situation, by applying these properties, it's possible to deal with the squared functions inside a square root cleanly. The Root Rule, in particular, allows us to move the limit inside the square root, which is crucial to finding that \( \lim_{x \to a} \sqrt{f^2(x) + g^2(x)} \) simplifies to \( \sqrt{10} \).

Calculus Problem Solving

Problem-solving in calculus usually involves a structured approach. Breaking down a problem into manageable steps can make seemingly hard tasks much more straightforward. Here’s a general method to tackle calculus problems:

• Understand the problem: Identify what you’re asked to find and the conditions given.
• Break down the expression: Simplify complicated expressions using algebra and known calculus rules.
• Apply limit rules appropriately: Use the properties of limits to compute or restructure the expression.
• Solve step-by-step: Arrive at the solution by using a logical sequence of computations.

In the exercise, following these steps made the computation of the limit of \( \sqrt{f^2(x) + g^2(x)} \) straightforward once the problem was understood. Calculus problems often reward patience and methodical checking of each part of the problem. Approach each step logically, and you will find the solution more manageable and accurate.