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find the limit. use l'hospital's rule if appropriate. if there is a more elementary method, consider using it. lim x→[infinity] x sin(3/x)

User Al Imran
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2 Answers

3 votes

Answer:

To find the limit of the function as x approaches infinity, we can use L'Hôpital's rule. Let's apply the rule:

lim x→∞ x sin(3/x)

We can rewrite this expression as:

lim x→∞ (sin(3/x))/(1/x)

Now, we can differentiate the numerator and denominator separately. Applying L'Hôpital's rule:

lim x→∞ (cos(3/x) * (-3/x^2))/(-1/x^2)

Simplifying further:

lim x→∞ (3cos(3/x))/1

Now, as x approaches infinity, the term 3cos(3/x) approaches 3cos(0) = 3.

Therefore, the limit is:

lim x→∞ (3cos(3/x))/1 = 3

So, the limit of x sin(3/x) as x approaches infinity is 3.

User Delevoye Guillaume
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To find the limit
\displaystyle\sf \lim_{{x\to\infty}} x\sin\left((3)/(x)\right), we can use L'Hôpital's rule.

Applying L'Hôpital's rule, we differentiate the numerator and the denominator separately. Let's start by differentiating the numerator.

Differentiating
\displaystyle\sf x with respect to
\displaystyle\sf x gives
\displaystyle\sf 1.

Now, let's differentiate the denominator.

Differentiating
\displaystyle\sf \sin\left((3)/(x)\right) with respect to
\displaystyle\sf x requires the chain rule. The derivative of
\displaystyle\sf \sin(u) with respect to
\displaystyle\sf u is
\displaystyle\sf \cos(u). So, the derivative of
\displaystyle\sf \sin\left((3)/(x)\right) with respect to
\displaystyle\sf x is
\displaystyle\sf \cos\left((3)/(x)\right) \cdot \left(-(3)/(x^(2))\right).

Taking the limit of
\displaystyle(1)/(\cos\left((3)/(x)\right)\cdot\left(-(3)/(x^(2))\right))} as
\displaystyle\sf x approaches infinity, we obtain
\displaystyle\sf 0.

Therefore,
\displaystyle\sf \lim_{{x\to\infty}} x\sin\left((3)/(x)\right) =0.

Note that in this case, we can also use an elementary method without L'Hôpital's rule. Since
\displaystyle\sf \lim_{{x\to\infty}} (3)/(x)=0, we can substitute
\displaystyle\sf u=(3)/(x) and rewrite the limit as
\displaystyle\sf \lim_{{u\to 0}} (3)/(u)\sin(u). As
\displaystyle\sf \lim_{{u\to 0}} (3)/(u)=+\infty and
\displaystyle\sf \lim_{{u\to 0}} \sin(u)=0, the limit is also
\displaystyle\sf 0.

User Tim Dunphy
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