Limits and Continuity


In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.


lim ((4―g(x)) / x ) = 1
x→0

Verified step by step guidance

1. First, understand that the problem involves finding the limit of a function as x approaches 0. The given limit statement is \( \lim_{{x \to 0}} \frac{{4 - g(x)}}{x} = 1 \).
2. Recognize that this is a limit problem involving a fraction where the numerator is \(4 - g(x)\) and the denominator is \(x\). As \(x\) approaches 0, the fraction approaches 1.
3. To solve for \(g(x)\), consider the form of the limit. For the limit to exist and equal 1, the numerator must approach 0 as \(x\) approaches 0. This implies that \(4 - g(x)\) must approach 0.
4. Set up the equation based on the above reasoning: \(4 - g(x) = 0\). Solve this equation to find \(g(x)\).
5. Conclude that for the limit to hold, \(g(x)\) must approach a specific value as \(x\) approaches 0. Solve the equation \(4 - g(x) = 0\) to find that value.