Problem 9 - Calculus: Early Transcendentals

Question

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. $$\lim _{x \rightarrow 1}(f(x)-g(x))$$

Step-by-Step Solution

Answer

Answer: The limit of the function (f(x) - g(x)) as x approaches 1 is equal to 5.

Step 1: Identify the given limits

We are given the limits of three functions as x approaches 1: 1. \(\lim _{x \rightarrow 1} f(x) = 8\) 2. \(\lim _{x \rightarrow 1} g(x) = 3\) 3. \(\lim _{x \rightarrow 1} h(x) = 2\)

Step 2: Apply the limit law for subtraction

We want to compute the limit of the function (f(x) - g(x)) as x approaches 1. According to the Limit Law for Difference: $$\lim _{x \rightarrow c}(f(x) - g(x)) = \lim _{x \rightarrow c} f(x) - \lim _{x \rightarrow c} g(x)$$ In our case, c = 1. Thus we have: $$\lim _{x \rightarrow 1}(f(x) - g(x)) = \lim _{x \rightarrow 1} f(x) - \lim _{x \rightarrow 1} g(x)$$

Step 3: Substitute the given limits

We already know the limits of f(x) and g(x) as x approaches 1, so we can substitute their values in the equation: $$\lim _{x \rightarrow 1}(f(x) - g(x)) = 8 - 3$$

Step 4: Calculate the final limit

Finally, perform the subtraction: $$\lim _{x \rightarrow 1}(f(x) - g(x)) = 5$$ Thus, the limit of the function (f(x) - g(x)) as x approaches 1 is equal to 5. The limit laws used in this computation are the Limit Law for Difference and substitution of known limits.

Key Concepts

Limit of a Function

Understanding the limit of a function is crucial in the exploration of calculus. In essence, the limit captures the behavior of a function as the input value approaches a certain point. For a limit to exist, a function must approach a specific value as the input gets increasingly close to a point from either side of the interval. The notation \(\lim _{x \rightarrow c} f(x)\) represents the limit of the function \(f(x)\) as \(x\) approaches \(c\).

When we say \(\lim _{x \rightarrow 1} f(x) = 8\), we imply that as \(x\) gets closer to 1, the value of \(f(x)\) gets closer to 8, regardless of whether \(f(x)\) actually equals 8 when \(x=1\). It's an essential concept because it leads to the understanding of function continuity, derivatives, and integrals—foundational concepts in calculus.

Difference of Limits

When dealing with the subtraction of functions, a fundamental limit law comes into play—the Limit Law for Difference. This law states that the limit of the difference between two functions is the difference of their limits, provided that these limits exist. Mathematically expressed as \(\lim _{x \rightarrow c}(f(x) - g(x)) = \lim _{x \rightarrow c} f(x) - \lim _{x \rightarrow c} g(x)\).

Applying this law in practical scenarios simplifies limit calculations. For example, if one needs to find \(\lim _{x \rightarrow 1}(f(x)-g(x))\), and has the individual limits of \(f(x)\) and \(g(x)\), one can just find the difference between these known limits. This is precisely what we did in the problem, where we found that as \(x\) approaches 1, the difference \(f(x) - g(x)\) approaches 5.

Limit Computation

Computation of limits is a systematic process involving a number of laws and properties of limits. Besides the Limit Law for Difference, there are other laws such as the laws of multiplication, division, and power of limits. While computing limits, it's often helpful to simplify the function if possible, use algebraic manipulation, or known values for common limits—like those of polynomial or trigonometric functions.

Direct substitution is one of the simplest methods when possible: If a function is continuous at \(c\), then \(\lim _{x \rightarrow c} f(x) = f(c)\). In the exercise provided, direct substitution couldn't be simpler. We know the limits as \(x\) approaches 1 for both \(f(x)\) and \(g(x)\), and by applying the Limit Law for Difference, we directly substitute to get our answer without the need for further complex calculations.