(Solved): 3) Prove the following limits by algebraic limit theorem, or any other theorem: a) \( \lim _{x \rig ...

3) Prove the following limits by algebraic limit theorem, or any other theorem: a) \( \lim _{x \rightarrow 1} \frac{2 x-1}{x^{2}+3} \) b) \( \lim _{x \rightarrow 0} \frac{\tan (x)}{x} \) c) \( \lim _{x \rightarrow 0} \frac{\cos (x)-1}{x} \) d) \( \lim _{x \rightarrow 0} \ln (x) \sin \left(\frac{1}{\ln (x)}\right) \) e) \( \lim _{x \rightarrow \infty}\left(\frac{x^{3}+2 x+5}{x^{3}+2 x+1}\right)^{2 x^{3}} \) f) \( \lim _{x \rightarrow \sin (3 x)} \frac{\sin (5 x)}{\sin (3 x)} \)