Question #320177

Evalute limits involving the by constructjng their respective table of values



1.lim t -> 0 t/(sin t)



2.lim t -> 0 (sin t)/(2t)•(1 - cot t)/t



3.lim t -> 0 (e ^ t - 1/t)




1
Expert's answer
2022-03-31T08:51:52-0400
  1. limt0tsint=limt0(sintt)1=1.\lim_{t\rightarrow0}\frac{t}{sin\,t}=\lim_{t\rightarrow0}(\frac{sin\,t}{t})^{-1}=1. It contains the well-known limit sintt\frac{sin\,t}{t} The values of function tsint\frac{t}{sin\,t} near are: t=0.05,tsint1.0004;t=0.05,\frac{t}{sin\,t}\approx1.0004; t=0.03,tsint1.0002;t=0.03,\frac{t}{sin\,t}\approx1.0002; t=0.01;tsint1.00002.t=0.01;\frac{t}{sin\,t}\approx1.00002. From the latter we can conclude that the limit is 1. Another way to compute the limit is to use L'Hôpital's rule.
  2. limt0sint2t1cottt=+\lim_{t\rightarrow0}\frac{sin\,t}{2t}\frac{1- cot\,t}{t}=+\infty . The first fraction is the well-known limit multiplied by 12\frac{1}{2}. The second fraction can be rewritten in the form: 1cottt=1costsintt=costsinttsint\frac{1-cot\,t}{t}=\frac{1-\frac{cos\,t}{sin\,t}}{t}=\frac{cos\,t-sin\,t}{t\,sint}. This fraction approaches infinity (the term costsintcos\,t-sin\,t approaches 1 and the term tsintt\,sint approaches 0). Another way to compute the limit is to use L'Hôpital's rule.
  3. limt0et1t=1\lim_{t\rightarrow0}\frac{e^t-1}{t}=1. It is the well-known known limit et1t\frac{e^t-1}{t} . The values of function et1t\frac{e^t-1}{t} near are: t=0.05t=0.05, et1t1.0152;\frac{e^t-1}{t}\approx1.0152; t=0.03,et1t1.0152;t=0.03,\frac{e^t-1}{t}\approx1.0152; t=0.01,t=0.01, et1t1.0050.\frac{e^t-1}{t}\approx1.0050. Another way to compute the limit is to use L'Hôpital's rule.

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