Find the limit (if it exists). (If an answer does not exist, write DNE and expla

Question

Find the limit (if it exists). (If an answer does not exist, write DNE and explain.) lim t --> 0+ of (3+e^(2t))i + [ (13e^(6t)?13)/9t ]j + e^(-9t)k . . . State the rule used to find the indicated limit and execute: lim t -->0+ of ((13e^6t-13)/9t) lim t--> 0+ = _________ . . . Write the above answer as an integer, fraction, or terminating decimal value = _________ . . . Finally, give the final limit of the vector-value function, at the beginning :lim t--> 0+ of (3+e^(2t))i + [ (13e^(6t) - 13)/9t ]j + e^(-9t)k =_____________

Explanation / Answer

lim t--> 0+ (3+e^(2t))i + [ (13e^(6t) - 13)/9t ]j + e^(-9t)k =[ 3+ e^(2 * 0)] i +lim t--> 0+ [ (13e^(6t) - 13)/9t ]j + e^(-9*0)k ; for evaluating lim t--> 0+ [ (13e^(6t) - 13)/9t ]j Using L-Hospitals Rule, we differentiate numerator and denominator with respect to t separately. which gives lim t--> 0+[ 13e^6t*6*1-0] /9*1 = 26/3 j .So the lim t--> 0+ of (3+e^(2t))i + [ (13e^(6t) - 13)/9t ]j + e^(-9t)k = 4i+ (26/3) j +k. NOTE: L-Hospitals Rule states that for functions f and g which are differentiable on I ?{c} , where I is an open interval containing c: If lim x---->c of f(x)= lim x---->c of g(x) = 0 or +/- infinity , and lim x------>c of f'(x)/g'(x) exists, and for all x in I with x ? c, then .The differentiation of the numerator and denominator often simplifies the quotient and/or converts it to a determinate form, allowing the limit to be evaluated more easily.

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