TITLE: $ \lim_{x \to \infty} \frac{ 5 + 7x + \sin(3x) } { (4+3 \sin x) e^{\sin(2x) } }$
QUESTION [0 upvotes]: So, I did the substitution $ x= \frac{1}{t}$ , using maclaurain after that I got the limit as $ \infty$ but how would I do it without maclurin? I know I can't use l'hopital because of the $ \sin$ function in the limit. So, would there be a way to do this using pure algebraic substitutions ?

REPLY [3 votes]: Hint: $\frac {5+7x+\sin (3x)} {(4+3 \sin x)e^{\sin x}} \geq \frac {7x-5-1} {7e^{1}} \to \infty$.

REPLY [2 votes]: When $x$ is large, the dominant term in the numerator is $x$ and in denominator it is an uncertain but positive finite number. Hence, the limit is $\infty$.

REPLY [0 votes]: Note that $\sin x$ is bounded between $-1$ and $1$, so all terms are bounded except $7x$. The denominator is positive, so the limit is $+\infty$.
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