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- Under what condition does eat converge to 0 as t->infinity? a = 0 a > 0 a < 0 a <= 0The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that g(x) = 0 for 4 distinct values of x in (a, b).What is the minimum number, k, of z in (a, b) such that g'(z) = 0? k=?1. Show (in terms of ε - ? ) that a function f : [2 ; 7]→ R be defined by f(x) = square root of (x2 +1) is uniformly continuous.
- Prove using the ϵ−n0 definition that the sequence Xn=(9−7n)/(8−13n) converges, and find its limit.The function g is continuous on the interval [a, b] and is differentiable on (a, b).Suppose that g(x) = 0 for 9 distinct values of x in (a, b).What is the minimum number, k, of z in (a, b) such that g'(z) = 0?k =true or false , prove your answer 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 ].
- 1. Use the definition of the limit ( epsolon - delta ) to show thatlim of 1/z as z approaches -i2. Give the condition which ensure that |ez| < 1 where z in C.Prove that the sum T in the Trapezoidal Rule for ∫ba ƒ(x) dx is aRiemann sum for ƒ continuous on 3a, b4 . (Hint: Use the IntermediateValue Theorem to show the existence of ck in the subinterval[xk-1, xk] satisfying ƒ(ck) = (ƒ(xk-1) + ƒ(xk))/2.)Suppose f is differentiable on an interval I and f'(x) >0for all numbers x in I except for a single number c. Provethat f is increasing on the entire interval I.
- Let S be a nonempty set of real numbers that is bounded from above (below) and let x = sup S (inf S). Prove that either x belongs to S or x is an accumulation point of S.Use the definition of convergence to prove lim {n→∞} (cos n)/(n2-n+1) = 0.Prove that if f: A to R is continuous at x=c, then for all (x_n) converging to c, it follows that f(x_n) converges to f(x).