Using L’Hopital’s rule or otherwise, find the limit lim 𝑥 → 0 ( 2 sin ⁡ ( 𝑥 ) + 2 sin ⁡ ( 2 𝑥 ) 3 sin ⁡ ( 𝑥 ) ) .单项选择题

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Question text Use l'Hopital's rule to calculate the limit [math: limx→∞4xln⁡(5x+1)]. Hint: You should rewrite the product as a fraction of the type [math: 0/0]. Answer: [input] Check Question 1

Using L’Hopital’s rule or otherwise, find the limit lim 𝑥 → 0 ( − 3 𝑥 2 + 2 𝑒 𝑥 − 2 − 11 𝑥 2 + 𝑥 ) .

Using L’Hopital’s rule or otherwise, find the limit lim 𝑥 → 0 ( − 2 𝑥 2 + 7 𝑒 𝑥 − 7 − 11 𝑥 2 + 2 𝑥 ) .

Are the following statements true or false? I. L'Hopital's Rule says that the limit of a function equals the limit of the derivative. [ Select ] False True II. L'Hopital's Rule can sometimes be used multiple times in the same problem. [ Select ] True False III. If lim 𝑥 → 𝑎 𝑓 ( 𝑥 ) = ∞ and   lim 𝑥 → 𝑎 𝑔 ( 𝑥 ) = ∞ , then   lim 𝑥 → 𝑎 ( 𝑓 ( 𝑥 ) − 𝑔 ( 𝑥 ) ) = 0  . [ Select ] True False IV. If lim 𝑥 → 𝑎 𝑓 ( 𝑥 ) = 0   and   lim 𝑥 → 𝑎 𝑔 ( 𝑥 ) = 0   , then   lim 𝑥 → 𝑎 ( 𝑓 ( 𝑥 ) − 𝑔 ( 𝑥 ) ) = 0  . [ Select ] False True V. If lim 𝑥 → 𝑎 𝑓 ( 𝑥 ) = ∞    and   lim 𝑥 → 𝑎 𝑔 ( 𝑥 ) = 0   , then   lim 𝑥 → 𝑎 ( 𝑓 ( 𝑥 ) − 𝑔 ( 𝑥 ) ) = ∞   . [ Select ] False True

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