1) Given the vector-valued function r(t)=⟨e−t,−et−t,tln⁡t⟩ a- Evaluate limt⇒0r(t) and Evaluate limt⇒∞r(t) b- Find r '(t) and evaluate r '(0) c- Find r''(t) d- Evaluate ∫r(t)dt

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1) Given the vector-valued function r(t)=⟨e−t,−et−t,tln⁡t⟩ a- Evaluate limt⇒0r(t) and Evaluate limt⇒∞r(t) b- Find r &#039;(t) and evaluate r &#039;(0) c- Find r&#039;&#039;(t) d- Evaluate ∫r(t)dt

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a- Evaluate the limitslimt⇒0r(t)=limt⇒0⟨e−t,−te−t,tln⁡t⟩=⟨limt⇒0e−t,limt⇒0−te−t,limt⇒0tln⁡t⟩=⟨e0,(0)e0,(0)ln⁡0⟩=⟨1,0,0⟩limt⇒∞r(t)=limt⇒∞⟨e−t,−te−t,tln⁡t⟩=⟨limt⇒∞e−t,limt⇒∞−te−t,limt⇒∞tln⁡t⟩=∞The limit does not exist because as tapproaches infinity, the limit of tIn(t) does not exist.b- Find r&#039;(t) and evaluate r&#039;(0)r(t)=ddt⟨e−t,−te−t,tln⁡t⟩=⟨ddt(e−t),ddt(−te−t),ddt(tln⁡t)⟩=⟨e−t,=e−t+te−t,ln⁡(t)+1⟩r′(t)=⟨−e−0,−e0+(0)−0,ln⁡(0)+1⟩=⟨−1,−1,∞⟩The value of r’(0) is undefinedc- Find r&#039;&#039;(t)We have r′(t)=⟨−e−t,e−t+te−t,ln⁡(t)+1⟩rt)=ddt⟨−e−t,e−t+te−t

1) Given the vector-valued function \r(t)=<< e^{-t},-et^{-t},t\ln t>> a- Evaluate

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