Suppose g(x) = {2x+1 if x≠0
5 if x=0.

Compute g(0) and lim x→0 g(x)

검증된 단계별 안내

Step 1: Identify the value of the function g(x) at x = 0. According to the piecewise definition, g(0) = 5.

Step 2: To find \( \lim_{x \to 0} g(x) \), consider the behavior of g(x) as x approaches 0 from both sides.

Step 3: For x ≠ 0, g(x) = 2x + 1. Analyze the limit \( \lim_{x \to 0} (2x + 1) \).

Step 4: Calculate \( \lim_{x \to 0} (2x + 1) \) by substituting x = 0 into the expression 2x + 1, which simplifies to 1.

Step 5: Compare the limit \( \lim_{x \to 0} g(x) = 1 \) with g(0) = 5 to determine if the function is continuous at x = 0.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, g(x) has two definitions: one for x not equal to 0 and another specifically for x equal to 0. Understanding how to evaluate piecewise functions is crucial for determining values at specific points and analyzing their behavior around those points.

Limits

The limit of a function describes the behavior of the function as the input approaches a certain value. In this question, we need to find lim x→0 g(x), which involves evaluating the function as x gets close to 0, but not equal to 0. This concept is fundamental in calculus for understanding continuity and the behavior of functions at points of interest.

Continuity

A function is continuous at a point if the limit as x approaches that point equals the function's value at that point. For g(x), we need to check if lim x→0 g(x) equals g(0). If they are equal, g(x) is continuous at x=0; if not, it indicates a discontinuity, which is an important aspect of function analysis in calculus.

Suppose g(x) = {2x+1 if x≠05 if x=0.Compute g(0) and lim x→0 g(x)