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- If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?Use L'Hospital's rule to find the limits of the following: a.) (ex-1-x2+x-1)/(x2-2x+1) as x approaches 1 b.) (ex+e-x-2)/(1-cos(2x)) as x approaches 0 c.) (3sin(4x))/5x as x approaches 0 d.) (x3+x2-2x)/(x-1) as x approaches 1C. Show directly from the definition (with epsilon - delta definition; without using Arithmetic of Limits or similar results) that f(x) = x3 + 1 is continuous at every real number c.
- Using the L’Hopital rule to evaluate the following limits. Show all steps and mathematical work. key- a) involves indeterminate quotients b) involves indeterminate products c) involves indeterminate powersFind the limit using the squeeze theorem. Assume the function g satifies the inequality 1≤g(x)≤sin2(x)+1 for x near 0.evaluate the limits using the limit laws A) lim (x^3 - 2x^2 +1) x is approaching -1 B) lim (t^2 - t)/(t+1) t is approaching 1 C) lim (1+cos(x))/(x^3 +2)
- Find the following limits. limx->+infinity (sin2x)/(x2+1) limx->+infinity (e-2x cosx) limx->+infinity (e-x+2cos3x)Find the limit x appraoaches infinity (e - 2x cos x)f(x) = { x2 x<= -2, 4 -2<x<0, 4+x3 x>=0 Which statements are true? (i) f(x) is differntiable at x = - 2 (ii) f(x) has a limit at x = - 2 (iii) f(x) is contionous at x = - 2
- i. Using Arithmetic of Limits, find limn→∞ 2n / (3n + 5).ii. Working directly from the definition of limits, give a direct verification that your answer in (i) is correct. (Your answer should involve the letter ε.)B. Using the definition of limit (so, without using Arithmetic of Limits), show thati. limn→∞ (4 + n) / 2n = 1/2ii. limn→∞ 2/n + 3/(n+1) = 0