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Concept explainers
Indeterminate Forms and I’Hopital’s Rule,
Let f and g be differentiable over an open interval containing
If
and if
exists, then
The forms
Since, for
Use this method to find the following limits. Be sure to check that the initial substitution results in an indeterminate from.
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Chapter 1 Solutions
Calculus and Its Applications (11th Edition)
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- Demostrate thatarrow_forwardA function is called ["", "", "", ""] on the open interval (�,�) continous. differentiable, linear , or defined? thanks for help aegkorwkgwrohiwk htwkhow tarrow_forward3 We want to sketch the function f(x) = 2 + x Note that f'(x) = State the domain of f(x). Domain: x-intercept(s): y-intercept(s): lim x → 3 18 + lim = x → -3x+18. Use algebra to determine any intercepts of f(x). Write the intercepts as ordered pairs. 9 --2 f(x) = Behavior on the right of the vertical asymptote: 2 2x + 3x9. x and f''(x) Locate any horizontal asymptote of f(x) by using limits. Write any horizontal asymptote as an equation. Horizontal Asymptote: f(x) is decreasing on State the location of the vertical asymptote of f(x) and use limits to describe the behavior of the function on each side of the asymptote. Location of Vertical Asymptote: Behavior on the left of the vertical asymptote: f(x) = 6 54 6x 54 x Use calculus to determine the interval(s) where f(x) is increasing and the intervals where f(x) is decreasing. f(x) is increasing onarrow_forward
- " (Sum Rule): Suppose f: ℝⁿ → ℝᵐ and g: ℝⁿ → ℝᵐ are functions, and let a ∈ ℝⁿ and b, c ∈ ℝᵐ be points. If lim(x→a) f(x) = b and lim(x→a) g(x) = c, then lim(x→a) (f(x) + g(x)) = b + c. Proof: Assume that lim(x→a) f(x) = b and lim(x→a) g(x) = c. Let ε > 0 be arbitrary. Then there exists δ₁ > 0 such that for x ∈ Dom(f) with d(x,a) < δ₁, we have ||f(x) - b|| < ε/2 (Equation 1.9). Similarly, there exists δ₂ > 0 such that for x ∈ Dom(g) with d(x,a) < δ₂, we have ||g(x) - c|| < ε/2 (Equation 1.10). Take δ := min(δ₁, δ₂) and let x ∈ Dom(f + g) satisfy d(x,a) < δ. Since x ∈ Dom(f) and d(x,a) < δ₁, Equation 1.9 holds. Furthermore, x ∈ Dom(g) and d(x,a) < δ₂, so Equation 1.10 applies. We can combine these inequalities: ||f(x) + g(x) - (b + c)|| = ||(f(x) - b) + (g(x) - c)|| ≤ ||f(x) - b|| + ||g(x) - c|| < ε/2 + ε/2 = ε. This shows that for all x ∈ Dom(f + g) with d(x,a) < δ, we have ||f(x) + g(x) - (b + c)|| < ε. Therefore, f(x) + g(x) → b + c as x → a." I…arrow_forwardThe second picture is and example problemarrow_forwardUsing the limit definition of the derivative of a function, set up the derivatives of the three functions listed below. Then, match the derivative each function you worked on with the correct limit from the list a), b), c), d), e), and f). (Some extra limits are provided in this list). 13 – x + h – a) lim h→0 1 sin x - 2 b) lim C) (2+ h)5 – 32 d) lim h→0 h cos ( + h) 2 e) lim h→0 h (2)5+h – 32 f) lim h→0 Here is the list of functions you need to work on and the space for choosing the limit that will comple the equation of the derivative: For f(x) = V3 – x, f'(x) Choose.. + • For f(æ) = x°, f'(2) = Choose.. + • For f(x) = sin x, f'(5) = Choose... +arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage