Finding Limits Graphically


Let f(x) = {3 - x , x < 2
2, x = 2
x/2, x > 2


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a. Find limx→2+ f(x), limx→2− f(x), and f(2).

Verified step by step guidance

1. To find \( \lim_{x \to 2^+} f(x) \), we need to consider the behavior of the function as x approaches 2 from the right. According to the piecewise definition, for \( x > 2 \), \( f(x) = \frac{x}{2} \). Substitute x = 2 into this expression to find the right-hand limit.
2. To find \( \lim_{x \to 2^-} f(x) \), we need to consider the behavior of the function as x approaches 2 from the left. According to the piecewise definition, for \( x < 2 \), \( f(x) = 3 - x \). Substitute x = 2 into this expression to find the left-hand limit.
3. To find \( f(2) \), we need to evaluate the function at x = 2. According to the piecewise definition, for \( x = 2 \), \( f(x) = 2 \).
4. Compare the values of \( \lim_{x \to 2^+} f(x) \), \( \lim_{x \to 2^-} f(x) \), and \( f(2) \) to determine if the function is continuous at x = 2. A function is continuous at a point if the left-hand limit, right-hand limit, and the function value at that point are all equal.
5. Summarize the findings: If \( \lim_{x \to 2^+} f(x) \) and \( \lim_{x \to 2^-} f(x) \) are equal, then the two-sided limit exists. If this limit is also equal to \( f(2) \), then the function is continuous at x = 2.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity. Limits can be approached from the left (denoted as lim x→c−) or from the right (lim x→c+), which is crucial for analyzing piecewise functions.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the given function f(x), different formulas apply for x values less than, equal to, or greater than 2. Understanding how to evaluate piecewise functions at specific points is essential for finding limits and function values accurately.

Continuity

Continuity at a point means that the limit of a function as it approaches that point from both sides equals the function's value at that point. For the function f(x) at x = 2, checking continuity involves comparing lim x→2+ f(x), lim x→2− f(x), and f(2). If these values are equal, the function is continuous at that point; otherwise, it is discontinuous.