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Ch. 2 - Limits
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 2 - LimitsProblema 52
Capitolo 2, Problema 52

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).

Guida verificata passo dopo passo
1
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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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.
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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.
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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.
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