Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.5.27

Capitolo 7, Problema 7.5.27

Use l’Hôpital’s rule to find the limits in Exercises 7–52.
27. lim (x → (π/2)^-) (x - π/2) sec x

Guida verificata passo dopo passo

Identify the limit expression: \(\lim_{x \to (\pi/2)^-} (x - \pi/2) \sec x\).

Check the form of the limit by substituting \(x = \pi/2\) from the left side: \((x - \pi/2)\) approaches 0, and \(\sec x = \frac{1}{\cos x}\) tends to \(\pm \infty\) because \(\cos(\pi/2) = 0\). This suggests an indeterminate form of type \(0 \cdot \infty\).

Rewrite the expression to apply l’Hôpital’s rule by converting the product into a quotient. For example, write it as \(\frac{x - \pi/2}{\cos x}\) because \(\sec x = \frac{1}{\cos x}\).

Now the limit becomes \(\lim_{x \to (\pi/2)^-} \frac{x - \pi/2}{\cos x}\), which is of the form \(\frac{0}{0}\), suitable for l’Hôpital’s rule.

Apply l’Hôpital’s rule by differentiating numerator and denominator separately: differentiate numerator \(\frac{d}{dx}(x - \pi/2) = 1\), and denominator \(\frac{d}{dx}(\cos x) = -\sin x\). Then evaluate the new limit \(\lim_{x \to (\pi/2)^-} \frac{1}{-\sin x}\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Limits and One-Sided Limits

A limit describes the value a function approaches as the input approaches a certain point. One-sided limits consider the approach from only one side, such as from the left (denoted by the minus sign). Understanding how to evaluate these limits is essential for analyzing behavior near points where the function may be undefined.

Video consigliato:

05:50

One-Sided Limits

l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems.

Video consigliato:

5:50

Power Rules

Behavior of Trigonometric Functions Near Critical Points

Trigonometric functions like sec(x) can have vertical asymptotes or undefined points at specific values, such as π/2 for sec(x). Understanding how these functions behave near such points helps in correctly applying limit techniques and interpreting the function’s behavior as x approaches these critical values.

Video consigliato:

Percorso guidato

6:04

Introduction to Trigonometric Functions

Pratica correlata

Domanda del libro di testo

43. Surrounding medium of unknown temperature A pan of warm water (46°C) was put in a refrigerator. Ten minutes later, the water’s temperature was 39°C; 10 min after that, it was 33°C. Use Newton’s Law of Cooling to estimate how cold the refrigerator was.

views

Domanda del libro di testo

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

views

Domanda del libro di testo

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.

8. y = ln kx, k constant

views

Domanda del libro di testo

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)