$$\mathop {Lim}\limits_{\theta \to 0} {{\sin \left( {\theta /2} \right)} \over \theta }\,\,\,$$ is

The following plot shows a function $$y$$ which varies linearly with $$x$$. The value of the integral $$\,\,{\rm I} = \int\limits_1^2 {y\,dx\,\,} $$

For the function $${e^{ - x}},$$ the linear approximation around $$x=2$$ is

For $$\left| x \right| < < 1,\,\cot \,h\left( x \right)\,\,\,$$ can be approximated as