Let fn(x) = x^n for x ∈ [0,1]. check if it is pointwise convergence. Define where it becomes discontinuous.

Q: Let xn -> 0 and (yn) be bounded. Prove xnyn -> 0.

A: A sequence is bounded if it is bounded above and below, that is to say, if there is a number, k,…

Q: Suppose X~N(196,20). Find x such that Pr(X > x) = 0.619.

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Q: Evaluate the following limit by first recognizing the sum as a Riemann sum for a function defined on…

A: The given  limit is limn→∞8n1n+2n+3n+....+nn

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A: Express the limit as a definite integral on the given interval.

Q: Suppose f(x) is continuous on [3, 8] and -5 < f'(x) < 3 for all x in (3, 8). Use the Mean Value…

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Q: For each natural number n and each number x in (-1,1), define f_n(x)=sqrt((x^2)+(1/n)) and define…

A: Uniformly continuity :- Let I⊂ℝ and for each n∈ℕ, let   fn : I →ℝ be  a function then the sequence…

Q: In your own words, explain why the divergence test is inconclusive when the limit of the sequence is…

A: Divergence test tests for divergence, not convergence.

Q: Let f_n(x)= nx / 1+nx^2.  Find the pointwise limit of (f_n) for all x in (0,infinity). Is the…

A: Given : fn(x)=nx1+nx2 .

Q: Define f(x)=x^3 for all x. Verify the ε-δ criterion for continuity at each point x_0.

A: X3 is continuous at every point in R.

Q: t fn(x) = sin^n(x) for all x [0,pi]. Prove that fn' is not uniformly convergent on [0,pi].…

A: We can prove it by showing that function is not pointwisw convergent.

Q: Evaluate the Riemann sum for f(x) = x³ - 6x on the interval [0, 3] taking 6 subintervals and using…

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Q: Express the limit as a definite integral on the given interval. [3(x)³ - 7x*]4x, 2 lim i = 1 dx…

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Q: The graph of f is shown in the figure. 6. 4 х 14 -4 -6+ 2-

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Q: Find  limit x 0  1/x^2if it exists.

A: We find the limit numerically

Q: k=1 diverge or converge? If the sequence converges, find the limit. *45. Show that if x is any real…

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Q: Use the MVT to prove that if f' (x) ≤ 2 for x &gt; 0 and f (0) = 4, then f (x) ≤ 2x + 4 for all x ≥…

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Q: Express as a definite integral the limit as x approaches infinite of 1/n[(1/n)^2 + (2/n)^2 +...+…

A: Step 1:

Q: Is there a sequence of continuous functions: fn : [0,1]-->R that is pointwise convergent but the…

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Q: H(3)=6 and the limit as x approaches 3 of H(x) is 6. Mustache be continuous at (3,6)? State why.

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