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- 15.2.65 #22 Use the method of your choice to evaluate the following limit. lim y ln y/x = (x,y)→(4,0)arrow_forwardFrom the definition of the intergal, we have lim n->infinity 2/n sigmma n k=1 (2+(4k/n)^3)=arrow_forwarda. Graph h(x) = x2 cos (1/x3) to estimate limx-->0 h(x), zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a proof.arrow_forward
- Use L' Hopital's to find the limit as x approaches positive infinity of: (e^(3x))/(x^(2)) I hope you have a nice day!arrow_forwardtrue or false prove your answer Suppose (an) has limit 4 and (bn) has limit L and that bn > an for infinitely manyvalues of n. Then L > 4.arrow_forwardH(3)=6 and the limit as x approaches 3 of H(x) is 6. Mustache be continuous at (3,6)? State why.arrow_forward
- Prove that Eulers constant e=(lim x->infinity symbol) [1+1/x]^xarrow_forwardThe continuous extension of (sin x)^x [0,π] a) Graph f(x)= (sin x)^x on the interval 0 ≤ x ≤ π. What value would you assign to f(x) to make it continuous at x = 0? b) Verify your conclusion in part (a) by finding limx→0+ f(x) with L’Hôpital’s Rule. c) Returning to the graph, estimate the maximum value of f(x) on [0, π].About where is max f(x) taken on? d) Sharpen your estimate in part (c) by graphing ƒ in the same window to see where its graph crosses the x-axis. To simplify your work, you might want to delete the exponential factor from the expression for f' and graph just the factor that has a zero.arrow_forwardAnswer both the questions. 1. Is the given statement true or false? Justify your answer. 'If f is continuous, decreasing function on [1, ∞) and lim x--> ∞ f(x) = 0, then f 1--> ∞ f(x) dx is convergent." 2. Calculate the third degree Taylor polynomial for y = sinh(x) around x = 1. (Hint: Use Euler's Formula)arrow_forward
- Determine whether the following statement is true or false. If this statement is true, explain why. If this statement is false, provide a counterexample. If lim n→∞a_n = π, then a_n is divergent.arrow_forward5. True or False? You do not need to prove your answers (but know how you would prove them!). d) Suppose that for every E > 0, the interval (4−E, 4 +E) contains infinitely many terms of (an). Then (an) converges to 4. e) If (an) is Cauchy and p is a number such that for all k ∈ N, |ak − p| < 1/2, then (an) converges to a limit L that lies in the interval [p − 1/2, p + 1/2]. f) If (an) is Cauchy and p is a number such that for infinitely many values of k, |ak − p| < 1/2, then (an) converges to a limit L that lies in the interval [p − 1/2, p + 1/2]. g) Suppose (an) has limit 4 and (bn) has limit L and that bn > an for infinitely many values of n. Then L > 4.arrow_forwardQ2)valuate the limit: Limit x approach to 2 [sin(pi.x)]/[x^2-x-2]arrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage