Problem 27 - Single Variable Calculus

Question

You are given that \(\lim _{x \rightarrow a} f(x)=2\), \(\lim _{x \rightarrow a a} g(x)=4\), and \(\lim _{x \rightarrow a} h(x)=-1 .\) Find the indicated limit. \(\lim _{x \rightarrow a}\left\\{[h(x)]^{2}-f(x) g(x)\right\\}\)

Step-by-Step Solution

Answer

The short answer for the given question is: \(\lim_{x \rightarrow a} [h(x)]^2 - f(x)g(x) = -7\)

Step 1: Identify Limit Properties

We need to recall the limit properties that can be applied to the sum, difference, product, and power of functions. Let's list them down. 1. The limit of a sum is the sum of the limits: \(\lim_{x \rightarrow a} ( f(x) + g(x) ) = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(composition(x))\) 2. The limit of a difference is the difference of the limits: \(\lim_{x \rightarrow a} ( f(x) - g(x) ) = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x)\) 3. The limit of a product is the product of the limits: \(\lim_{x \rightarrow a} ( f(x) . g(x) ) = \lim_{x \rightarrow a} f(x) . \lim_{x \rightarrow a} g(x)\) 4. The limit of a function raised to a power is the limit of the function raised to that power: \(\lim_{x \rightarrow a} f(x))^n = (\lim_{x \rightarrow a} f(x))^n\) Now, let's apply these properties to the given expression.

Step 2: Apply Limit Properties on the Expression

We are asked to find the limit of the expression \(\lim_{x \rightarrow a} [h(x)]^2 - f(x)g(x)\). Using the limit properties mentioned above, we can break down the expression as follows: \(\lim_{x \rightarrow a} [h(x)]^2 - f(x)g(x) = \lim_{x \rightarrow a} [h(x)]^2 - \lim_{x \rightarrow a} f(x)g(x)\) Now, we can further break down each limit using property 3 for the product of functions, and property 4 for the power of a function. \(\lim_{x \rightarrow a} [h(x)]^2 = ([\lim_{x \rightarrow a} h(x)])^2\) \(\lim_{x \rightarrow a} f(x)g(x) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x)\)

Step 3: Substitute the Given Limits

We are given that: \(\lim_{x \rightarrow a} f(x) = 2\) \(\lim_{x \rightarrow a} g(x) = 4\) \(\lim_{x \rightarrow a} h(x) = -1\) Now, we can substitute these values into the broken down expression: \(([\lim_{x \rightarrow a} h(x)])^2 - \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x) = (-1)^2 - (2)(4)\)

Step 4: Compute the Final Limit Value

Now that we have the substituted values, we can easily compute the limit value: \((-1)^2 - (2)(4) = 1 - 8 = -7\) Thus, the final result is: \(\lim_{x \rightarrow a} [h(x)]^2 - f(x)g(x) = -7\)

Key Concepts

Limit of a Sum

Understanding the limit of a sum is crucial for simplifying complex expressions involving limits. When you add two functions together, the limit of the resulting function as it approaches a certain value is simply the sum of the limits of the individual functions as they approach that value. This is expressed as: \( \lim_{x \rightarrow a} ( f(x) + g(x) ) = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x) \). This property plays a foundational role in calculus and is useful when breaking down larger problems into manageable pieces. For example, if you have two functions, say \( m(x) \) and \( n(x) \), approaching values \( L \) and \( M \) respectively as \( x \) approaches \( a \), the sum \( m(x) + n(x) \) will approach \( L+M \). This makes calculating the limit of a combined function simple and straightforward.

Limit of a Difference

Just like the limit of a sum, the limit of a difference follows a similar logic. When you subtract one function from another, the limit of the resulting function as it nears a value is the difference of the limits of each of the original functions. This is written as: \( \lim_{x \rightarrow a} ( f(x) - g(x) ) = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x) \). This property allows you to decompose and evaluate more complex expressions with ease. For instance, if \( p(x) \) and \( q(x) \) approach limits \( P \) and \( Q \), respectively, as \( x \rightarrow a \), then the expression \( p(x) - q(x) \) will have a limit of \( P - Q \) as \( x \rightarrow a \). It's a straightforward and reliable way to handle differences in limits without a fuss.

Limit of a Product

The limit of a product is a property that becomes handy when you're dealing with the multiplication of functions. According to this property, the limit of the product of two functions is the product of their limits. Mathematically, it is expressed as: \( \lim_{x \rightarrow a} ( f(x) \cdot g(x) ) = \lim_{x \rightarrow a} f(x) \cdot \lim_{x \rightarrow a} g(x) \). This property simplifies the analysis of functions that are multiplied together by allowing you to analyze each of their limits separately first. For example, if \( r(x) \) and \( s(x) \) have limits \( R \) and \( S \) as \( x \rightarrow a \), then the product \( r(x) \cdot s(x) \) will have a limit of \( R \cdot S \). This is particularly useful when dealing with polynomial and rational expressions.

Limit of a Power

The limit of a power is essential when you are working with functions raised to an exponent. It tells us that the limit of a function raised to a power is the limit of the function itself, raised to that power. This can be represented as: \( \lim_{x \rightarrow a} [f(x)]^n = \left( \lim_{x \rightarrow a} f(x) \right)^n \). This property makes it easier to evaluate the limits of functions with exponents by focusing initially on the base function. For example, if a function \( t(x) \) approaches the limit \( T \) as \( x \rightarrow a \), then \([t(x)]^n\) will reach the value \( T^n \). This is especially handy for polynomial expressions or any function that naturally includes exponents, providing a streamlined process for tackling those problems.