avissidep

2021-10-17

Use continuity to evaluate the limit. $\underset{x\to \pi }{lim}\mathrm{sin}\left(x+\mathrm{sin}x\right)$

au4gsf

x and $\mathrm{sin}x$ are continuous, therefore $x+\mathrm{sin}x$ is continuous
Recall that : f(g(x)) is continuous if f(x) and g(x) are continuous
Therefore $\mathrm{sin}\left(x+\mathrm{sin}x\right)$ is a continuous function
Recall that : If f(x) is continuous at x=a, then $\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$
Therefore
$\underset{x\to \pi }{lim}\mathrm{sin}\left(x+\mathrm{sin}x\right)=\mathrm{sin}\left(\pi +\mathrm{sin}\pi \right)=\mathrm{sin}\left(\pi +0\right)=\mathrm{sin}\pi =0$
Result:
$\underset{x\to \pi }{lim}\mathrm{sin}\left(x+\mathrm{sin}x\right)=0$

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