Assume lim x→1 f(x)=8,lim x→1 g(x)=3, and lim x→1 h(x)=2 Compute the following limits and state the limit laws used to justify your computations.


lim x→1 (f(x)−g(x))

Verified step by step guidance

1. Identify the given limits: \( \lim_{{x \to 1}} f(x) = 8 \) and \( \lim_{{x \to 1}} g(x) = 3 \).
2. Recognize that the problem asks for \( \lim_{{x \to 1}} (f(x) - g(x)) \).
3. Apply the limit law for the difference of two functions: \( \lim_{{x \to a}} (f(x) - g(x)) = \lim_{{x \to a}} f(x) - \lim_{{x \to a}} g(x) \).
4. Substitute the known limits into the equation: \( \lim_{{x \to 1}} f(x) - \lim_{{x \to 1}} g(x) = 8 - 3 \).
5. Conclude that the limit is the result of the subtraction: \( 8 - 3 \).

Key Concepts

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In this case, we are interested in the behavior of the functions f(x), g(x), and h(x) as x approaches 1. Understanding limits is fundamental in calculus as it lays the groundwork for continuity, derivatives, and integrals.

Limit Laws

Limit laws are a set of rules that allow us to compute limits of functions based on the limits of their components. For example, the limit of the difference of two functions is the difference of their limits, which is expressed as lim x→c (f(x) - g(x)) = lim x→c f(x) - lim x→c g(x). These laws simplify the process of finding limits and are essential for solving limit problems.

Difference of Limits

The difference of limits states that if the limits of two functions exist as x approaches a certain value, then the limit of their difference also exists. Specifically, if lim x→c f(x) = L and lim x→c g(x) = M, then lim x→c (f(x) - g(x)) = L - M. This concept is crucial for solving the given limit problem involving f(x) and g(x).