Concept explainers
Evaluating a Limit In Exercises 3 and 4, use Example 1 as a model to evaluate the limit
Over the region bounded by the graphs of the equations.
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Calculus: Early Transcendental Functions
- Use Example 1 as a model to evaluate the limit lim n→∞ n f(ci)Δxi i = 1 over the region bounded by the graphs of the equations. (Round your answer to three decimal places.) f(x) = x , y = 0, x = 0, x = 6 Hint: Let ci = 6i2 n2 .arrow_forwardComputing a limit graphically and analyticallyarrow_forwardLimit and Continuity In Exercises , find the limit (if it exists) and discuss the continuity of the function. 14. lim (x, y)→(1, 1) (xy) /(x^2 − y^2 ) 16. lim (x, y)→(0, 0) (x^2 y) /(x^4 + y^2)arrow_forward
- Advanced Calculus: Use the Bolzano–Weierstrass Theorem to prove that if f is a continuous function on [a,b], then f is bounded on [a,b] (that is, there exists M > 0 such that |f(x)| ≤ M for all x ∈[a,b]). (Hint: Give a proof by contradiction.)arrow_forwarda) does the function show symmetry? b) Calculates all boundary behaviors of the function, that is, all behaviors at the edges of the definition area.arrow_forwardDefinition: The AREA A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles PS: For (a) it is only choosing one of the right selection to determine.arrow_forward
- (Term-by-term Differentiability Theorem). Let fn be differentiable functions defined on an interval A, and assume ∞ n=1 fn(x) converges uniformly to a limit g(x) on A. If there exists a point x0 ∈ [a, b] where ∞ n=1 fn(x0) converges, then the series ∞ n=1 fn(x) converges uniformly to a differentiable function f(x) satisfying f(x) = g(x) on A. In other words, Proof. Apply the stronger form of the Differentiable Limit Theorem (Theorem6.3.3) to the partial sums sk = f1 + f2 + · · · + fk. Observe that Theorem 5.2.4 implies that sk = f1 + f2 + · · · + fk . In the vocabulary of infinite series, the Cauchy Criterion takes the followingform.arrow_forwarda) Determien the domain and range: f(x,y) = (y -x) / [log(y) - log(x)] b) Use Hopital to show that the limit limx->2f(x, 2) exists. c) Show that the limit limx->0f(x, 2) exists.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
- (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_forwardEvaluate the limit using L'Hospital's Rule The limit of ((e^x) + 2x-1) / (2x) as x approaches zeroarrow_forwardLet f(x, y) =x2(y−1)x4+(y−1)2. Determine the existence of limit lim(x,y)→(0,1)f(x, y).arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage