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- Given that lim x→3 (5x − 5) = 10, illustrate Definition 2 by finding values of ? that correspond to ? = 0.1, ? = 0.05, and ? = 0.01.Let ?:ℝ2→ℝ and suppose that lim(?,?)→(3,0)? (?,?)=−10.Given that lim x→1 (4x − 3) = 1, illustrate Definition 2 by finding values of δ that correspond to ε = 0.1, ε = 0.05, and ε = 0.01. ε = 0.1 δ ≤ 1 ε = 0.05 δ ≤ 2 ε = 0.01 δ ≤ 3
- 8. Identify the values of c for which lim ?→? ?(?) exists.Given that lim x→4 (4x − 13) = 3, illustrate Definition 2 by finding values of delta that correspond to epsilon = 0.1, epsilon = 0.05, and epsilon = 0.01. epsilon = 0.1 delta≤ epsilon = 0.05 delta≤ epsilon = 0.01 delta≤6. Evaluate the following limitsa. lim?→9 3−√9 / 9−?b. lim?→0 √1+?−√1−? / ?c. lim?→−2 ?+2 / ?2+?−2
- How many values of 'a' are there such that lim h→0 ((sin^(−1) (a+h) − sin^(−1) (a)) / h) = 4?Prove the statement using the ?, ? definition of a limit. lim x→4 x2 − 2x − 8 x − 4 = 6 Given ? > 0, we need ? such that if 0 < |x − 4| < , then x2 − 2x − 8 x − 4 − 6 . We have x2 − 2x − 8 x − 4 − 6 < ? ⇔ (x − 4)(x + 2) x − 4 − 6 < ? ⇔ |x + 2 − 6| < ? [x ≠ 4] ⇔ |x − 4| < ?. Choose ? = . Then 0 < |x − 4| < ? ⇒ |x − 4| < ⇒ x2 − 2x − 8 x − 4 − 6 By the definition of a limit, lim x→4 x2 − 2x − 8 x − 4 = 6. View Picture of Questionfind the limit: lim ?(?) ?→∞ ?(?) = (6?3−7)/ (5?3+?2+1)