Textbook Question
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.37
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
37. lim (y → 0) (√(5y + 25) - 5) / y
Verified step by step guidance1
First, identify the form of the limit as \( y \to 0 \) for the expression \( \frac{\sqrt{5y + 25} - 5}{y} \). Substitute \( y = 0 \) to check if it results in an indeterminate form.
Since substituting \( y = 0 \) gives \( \frac{\sqrt{25} - 5}{0} = \frac{5 - 5}{0} = \frac{0}{0} \), which is an indeterminate form, we can apply l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \( y \). The numerator is \( \sqrt{5y + 25} - 5 \), so find its derivative:
The derivative of the numerator is \( \frac{d}{dy} \left( \sqrt{5y + 25} - 5 \right) = \frac{1}{2\sqrt{5y + 25}} \times 5 = \frac{5}{2\sqrt{5y + 25}} \). The derivative of the denominator \( y \) is 1.
Rewrite the limit using these derivatives: \( \lim_{y \to 0} \frac{5}{2\sqrt{5y + 25}} \). Then, evaluate this new limit by substituting \( y = 0 \).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits and Indeterminate Forms
Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in expressions like 0/0 or ∞/∞, these are called indeterminate forms, requiring special techniques to evaluate the limit.
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One-Sided Limits
l’Hôpital’s Rule
l’Hôpital’s Rule is a method for evaluating limits that yield indeterminate forms 0/0 or ∞/∞ by differentiating the numerator and denominator separately and then taking the limit of their quotient.
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Differentiation of Composite Functions
To apply l’Hôpital’s Rule, one must differentiate functions, including composite ones like square roots. This involves using the chain rule, which differentiates the outer function and multiplies by the derivative of the inner function.
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Related Practice
Textbook Question
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Textbook Question
Evaluate the integrals in Exercises 111–114.
112. ∫₁^(eˣ) (1 / t) dt
Textbook Question
In Exercises 9 and 10, use implicit differentiation to find dy/dx.
9. y^e^x = x^y + 1
Textbook Question
7. Let A(t) be the area of the region in the first quadrant enclosed by the coordinate axes, the curve y=e^(-x), and the vertical line x=t, t>0. Let V(t) be the volume of the solid generated by revolving the region about the x-axis. Find the following limits.
a. lim(x→∞)A(t)
1
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