State the value of the limit, if it exists lim x→0 sin(x) multiplied with 3x^3+2x^2/x^2 Anybody? :)

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State the value of the limit, if it exists lim x→0 sin(x) multiplied with 3x^3+2x^2/x^2 Anybody? :)

Calculus1
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ok this is a big one
so stay with me
first we separate the lim

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\[\lim_{x \rightarrow 0} (3x+2) \times \lim_{x \rightarrow 0}(\frac{ \sin(x) }{ x ^{2} })\]
i forgot to times x^2 with sin(x)
ok
\[\lim_{x \rightarrow 0}(\frac{ \sin(x)x ^{2} }{ x ^{2} }) \times \lim_{x \rightarrow 0} (3x+2)\]
ur following the steps right
wow! :)
Yep! Think so…
3x+2 is a polynomial and thus everywhere continuous so \[\lim_{x \rightarrow 0} (3x+2)= 2+3(0)=2\]
and the \[\lim_{x \rightarrow 0} \frac{ \sin(0) \times 0 }{ 0 } =0\]
\[ 2 \times 0=0\]
got it
if u have similar denominator like x^2 in ur case u can separate the limit
Yup! Think so…. :)
I have to go through it a few times to make sure I understand everything :)
Thank you so much!
Any possibilities of some follow-up questions later on?
Yea sure
you simply can not say this: \[\lim_{x \rightarrow 0}(\frac{ \sin(x)x ^{2} }{ x ^{2} }) \] \[\implies \lim_{x \rightarrow 0} \frac{ \sin(0) \times 0 }{ 0 } =0\] the whole point of this is that dividing by zero you cannot do, so you check what the function in question does as you approach very close to zero. so: \[\lim\limits_{ x→0} \, \, \sin(x) .\frac{3x^3+2x^2}{x^2}\] \[=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{3x^3+2x^2}{x^2}\] \[=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{\frac{3x^3}{x^2}+\frac{2x^2}{x^2}}{\frac{x^2}{x^2}}\] \[=\lim\limits_{ x→0} \, \, \sin(x) .\lim\limits_{ x→0}\frac{3x+2}{1}\] \[=0 \times 2 = 0\]
Thank you so much for your reply! :)

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