4. Find the limit of the variable quantity $x=\frac{a z+1}{z}$ as $z \rightarrow \infty$.
Solution. Transform the variable by dividing all terms of the numerator by the denominator. We get: $x=a+\frac{1}{z} ; x-a=\frac{1}{z}$.

Notice that the larger $z$ is, the closer the values of the variable $x$ are to the constant $a$, since the condition $|x-a|<\varepsilon$ is satisfied, where $\mathbf{\varepsilon}-$ is an arbitrarily small quantity. Therefore, $\lim _{z \rightarrow \infty} x=a$.