Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. h(t)=2t−3t21+t3,a=1

Question

Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a.  h(t)=2t−3t21+t3,a=1

Faiza Fuller · Accepted Answer

Given:&amp;nbsp;h(t)=2t−3t21+t3,a=1&amp;nbsp;Definition of continuity:&amp;nbsp;If the conditions below are met, a real-valued function f(x) is said to be continuous at x=a.1.&amp;nbsp;limx→af(x)&amp;nbsp;exists.&amp;nbsp;2. f(a) is defined&amp;nbsp;3.&amp;nbsp;limx→af(x)=f(a)&amp;nbsp;1. Consider,&amp;nbsp;limt→1h(t)=limt→1[2t−3t21+t3]&amp;nbsp;Use the property of limit that,&amp;nbsp;limx→af(x)g(x)=f(a)g(a), where&amp;nbsp;g(a)≠0&amp;nbsp;Here,&amp;nbsp;f(t)=2t−3t2&amp;nbsp;and&amp;nbsp;g(t)=1+t3&amp;nbsp;⇒f(1)=2(1)−3(1)2=−1&amp;nbsp;and&amp;nbsp;g(1)=1+(1)3=2≠0&amp;nbsp;Therefore,&amp;nbsp;limt→1[2t−3t21+t3]=limt→1[f(t)g(t)]&amp;nbsp;=f(1)g(1)&amp;nbsp;=−12&amp;nbsp;limt→1[2t−3t21+t3]=−12&amp;nbsp;Thus,&amp;nbsp;limt→1h(t)&amp;nbsp;exists.&amp;nbsp;2.Find h(1):&amp;nbsp;h(1)=2(1)−3(1)21+(1)3&amp;nbsp;=2−31+1&amp;nbsp;=−12&amp;nbsp;h(1)=−12&amp;nbsp;3. And from calculations,&amp;nbsp;limt→1h(t)=h(1)&amp;nbsp;As a result, all continuity requirements are met.Therefore, the function&amp;nbsp;&amp;lt;

Jeffrey Jordon · Accepted Answer

Answer is given below (on video)

Use the definition of continuity and the properties of limits to show that the f

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New Questions in Calculus 1