Calculate the following limits, or explain why they diverge to ∞, too, or otherwise. You may use any theorems we have proved in class or on homework. (a) lim n³ n∞ (2n+1)(3n+2) (4n+3)
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- ) lim (Vx+3 +Vx) 14mathematically before reading the proof. (It might be helpful to read the discussion that follows Theorem 2.4 or to look at Appendix A.) THEOREM 3.2 THEIS a function that is defined on I except possibly at the point c. Let I be an open interval that contains the point c and suppose a) The function f has limit L at c if and only if for each sequence {Xn) in I\ {c} that converges to c, the sequence (f (x,)} converges to L. b) Suppose that there are sequences (x,} and (y,} in I\ {c} that converge to c such that {ƒ(xn)} converges to L, and (f(yn)} converges to L2. If Li + L2, then the functionf does not have a limit at c. Proof Let P be the statement "the function f has limit L at c" and let Q be the statement "for each sequence {xn} in I\ {c} that converges to c, the sequence (f(xn)} converges to L". A proof that P Q will be left as an exercise. To prove that Q = P, we will prove the contrapositive. Suppose lim f (x) # L. By the negation of the definition, there exists e>0 such that for…a) Evaluate 19x(x² – 3x + 2 lim (Зх — 3)