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Sketching graphs Sketch a possible graph of a function f, together with vertical asymptotes, satisfying all the following conditions on [0, 4].
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Calculus: Early Transcendentals (3rd Edition)
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University Calculus: Early Transcendentals (3rd Edition)
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- Vertical and horizontal asymptotes Find all vertical andhorizontal asymptotes of the following functions.arrow_forwardCalculus 3 Functions of Several Variables; Limits and Continuity in Higher Dimensions Question 3: Read Example 5 and the boxed text “Two-Path Test for Nonexistence of a Limit” (p. 818 – 819). Explain what the two-path test says and how this shows that the limit in this example does not exist at the origin. Include the details involved in this particular example.arrow_forwardFinding a function with vertical asymptotes Find polynomials pand q such that f = p/q is undefined at 1 and 2, but f has a verti-cal asymptote only at 2. Sketch a graph of your function.arrow_forward
- Asymptotes Use analytical methods and/or a graphing utilityto identify the vertical asymptotes (if any) of the following functions.arrow_forwardSketch the graph of a function f which incorporates all the limit & derivative information below: (Be sure to include any asymptotes in your sketch of the graph f) • lim f(x)=0 lim f(x)=−∞ lim f(x)=∞ limf(x)=−∞ lim f(x)=1 • The function values of f, and its first derivative f′(x), and its second derivative, f′′(x), are undefined at x = −2 and at x = 0, and are defined for all other real number values of x. • The first derivative f′(x) is negative on the x-intervals (−∞, −2) and (−2, 0), is positive on the interval (0, 2), is zero at x = 2, and is negative on the interval (2, ∞). • The second derivative f′′(x) is negative on the x-interval (−∞,−2), positive on the interval (−2, −1), has value zero at x = −1, is negative on the intervals (−1, 0) and (0, 3), has value zero at x = 3, and is positive on the interval (3, ∞).arrow_forwardCalculus (Limits at Infinity; Horizontal asymptotes)arrow_forward
- True or False. If a statement below must always be true, write True and give the brief justification . Otherwise, write False, and give an example in which the statement is not true. Your example may be a graph. a. If lim x->a+ f(x)= lim x->a- f(x), then f(x) is continuous at x=a b. If f(2) = 6 and f"(2)= 8, then f(x) is increasing at x=2arrow_forwardSketching a function Sketch a graph of a function f that is continuouson (- ∞, ∞) and satisfies the following conditions.1. f′ > 0 on (- ∞, 0), (4, 6), and (6, ∞).2. f′ < 0 on (0, 4).3. f′(0) is undefined.4. f′(4) = f′(6) = 0arrow_forwardUniqueness of limits Show that a function cannot have two differentlimits at the same point. That is, if limxSc ƒ(x) = L1 andlimxSc ƒ(x) = L2, then L1 = L2.arrow_forward
- Graphs of piecewise functions Write a definition of the func-tions whose graphs are given.arrow_forward(Right and Left Limits). Introductory calculus coursestypically refer to the right-hand limit of a function as the limit obtained by“letting x approach a from the right-hand side.” (a) Give a proper definition in the style of Definition 4.2.1 ((Functional Limit).for the right-hand and left-hand limit statements: limx→a+f(x) = L and limx→a−f(x) = M. (b) Prove that limx→a f(x) = L if and only if both the right and left-handlimits equal L.arrow_forwardDefinition of infinite limit: Let X⊆ R, f: X -> R and a∈ X'. If for every M>0 there exists delta > 0 such that |f(x)| > M whenever x∈X and 0< |x-a| < delta then we say that the limit as x approaches a of f(x) is ∞ which is denoted as lim {x-> a} f(x) = ∞. Suppose a∈R, ∈>0, and f,g : N*(a,∈) ->R. If lim {x-> a} f(x) = L>0 and lim {x-> a} g(x)= ∞, prove lim {x-> a} (fg)(x)=∞.arrow_forward
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