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Evaluating limits graphically Sketch a graph of f and use it to make a conjecture about the values of
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Chapter 2 Solutions
Calculus, Single Variable: Early Transcendentals (3rd Edition)
- (Question pertaining to indeterminate limits) It is not uncommon for people to write: lim x approaches a f(x) = 0/0 a) Why is this not correct? b) Is 0/0 a number? No. Explain what 0/0 means in terms of the numerator and the denominator.arrow_forwardSketching functionsa. Sketch the graph of a function that is not continuous at 1, but isdefined at 1.b. Sketch the graph of a function that is not continuous at 1, buthas a limit at 1.arrow_forward(x^3-x^2) divided by (x-1) lim x-> 1 Evaluate the limit.arrow_forward
- Definition 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_forwardlim x → 9− f(x) = 2 and lim x → 9+ f(x) = 4. As x approaches 9 from the right, f(x) approaches 2. As x approaches 9 from the left, f(x) approaches 4. As x approaches 9 from the left, f(x) approaches 2. As x approaches 9 from the right, f(x) approaches 4. As x approaches 9, f(x) approaches 4, but f(9) = 2. As x approaches 9, f(x) approaches 2, but f(9) = 4. In this situation is it possible that lim x → 9 f(x) exists? Explain. Yes, f(x) could have a hole at (9, 2) and be defined such that f(9) = 4. Yes, f(x) could have a hole at (9, 4) and be defined such that f(9) = 2. Yes, if f(x) has a vertical asymptote at x = 9, it can be defined such that lim x→9− f(x) = 2, lim x→9+ f(x) = 4, and lim x→9 f(x) exists. No, lim x→9 f(x) cannot exist if lim x→9− f(x) ≠ lim x→9+ f(x).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_forward
- Finding limits from a graph Use the graph of f in the figure to find the following values or state that they do not exist. If a limit does not exist, explain why.arrow_forwardLim x -> 2 (x^2-1/x-2) The lim x --> 2 is both negative and positive. i dont remember how it do it at allarrow_forward. Analyzing infinite limits graphically The graph of h in thefigure has vertical asymptotes at x = -2 and x = 3. Analyzethe following limits.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_forwardlim x->0 ((x+k)2 -x2/k)arrow_forward‘’Limits of Algebraic Function Using Tables and Graphs’’. In short simple phrases or sentences, explain the process on how to determine the limit of an algebraic function using Tables and Graphs.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningCollege AlgebraAlgebraISBN:9781305115545Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning