avissidep

2021-10-17

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

au4gsf

Skilled2021-10-18Added 95 answers

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$\mathrm{sin}(x+\mathrm{sin}x)$ 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

$\underset{x\to \pi}{lim}\mathrm{sin}(x+\mathrm{sin}x)=\mathrm{sin}(\pi +\mathrm{sin}\pi )=\mathrm{sin}(\pi +0)=\mathrm{sin}\pi =0$

Result:

$\underset{x\to \pi}{lim}\mathrm{sin}(x+\mathrm{sin}x)=0$

Recall that : f(g(x)) is continuous if f(x) and g(x) are continuous

Therefore

Recall that : If f(x) is continuous at x=a, then

Therefore

Result: