Determine the following limits.

Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).

검증된 단계별 안내

Step 1: Understand the inequality constraint on g(x), which is 1 \(\leq\) g(x) \(\leq\) \(\sin\)^2(x) + 1 for all x near 0.

Step 2: Evaluate the behavior of \(\sin\)^2(x) as x approaches 0. Recall that \(\sin\)(x) \(\approx\) x when x is near 0, so \(\sin\)^2(x) \(\approx\) x^2.

Step 3: Determine the limit of \(\sin\)^2(x) + 1 as x approaches 0. Since \(\sin\)^2(x) \(\to\) 0 as x \(\to\) 0, it follows that \(\sin\)^2(x) + 1 \(\to\) 1.

Step 4: Apply the Squeeze Theorem. Since 1 \(\leq\) g(x) \(\leq\) \(\sin\)^2(x) + 1 and both bounds approach 1 as x approaches 0, the Squeeze Theorem implies that \(\lim\)_{x \(\to\) 0} g(x) = 1.

Step 5: Conclude that the limit of g(x) as x approaches 0 is 1, based on the application of the Squeeze Theorem.

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주요 개념

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

Limit of a Function

The limit of a function describes the value that the function approaches as the input approaches a certain point. In this case, we are interested in the behavior of g(x) as x approaches 0. Understanding limits is fundamental in calculus, as it lays the groundwork for concepts such as continuity and derivatives.

Squeeze Theorem

The Squeeze Theorem is a principle used to find limits of functions that are bounded by two other functions whose limits are known. If a function g(x) is squeezed between two functions that both approach the same limit as x approaches a certain value, then g(x) must also approach that limit. This theorem is particularly useful when direct evaluation of the limit is difficult.

Behavior of Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, exhibit periodic behavior and have specific limits as their arguments approach certain values. For example, sin^2(x) approaches 0 as x approaches 0. Understanding the behavior of these functions near critical points is essential for evaluating limits involving trigonometric expressions.

Determine the following limits.Assume the function g satisfies the inequality 1≤g(x) ≤sin^2 x + 1, for all values of x near 0. Find lim x→0 g(x).