2.1.44 Calculation of limits

Condition: a. Find the limit: \[ \lim _{x \rightarrow 0} \ln (x+1) \sin \left(\frac{1}{x}\right) \] b. Use the definition of limits to prove that if \[ \lim _{x \rightarrow a^{-}} f(x)=L \text { and } \lim _{x \rightarrow a^{-}} f(x)=M \text { then } L=M \text {. } \] c. Use the definition of limits to prove that if \[ \lim _{x \rightarrow \infty} g(x)=\infty \text { and } \lim _{x \rightarrow \infty} f(x)=L \text { then } \lim _{x \rightarrow \infty} f(g(x))=L \text {. } \]