The given limit is
\[ \lim_{x \rightarrow 0} \left( \frac{ax + b}{cx} + 1 \right). \]
Combine the terms inside the bracket by taking a common denominator:
\[ \frac{ax + b}{cx} + 1 = \frac{ax + b}{cx} + \frac{cx}{cx}. \]
\[ = \frac{ax + b + cx}{cx}. \]
\[ = \frac{(a + c)x + b}{cx}. \]
Split the fraction:
\[ \frac{(a + c)x}{cx} + \frac{b}{cx}. \]
\[ = \frac{a + c}{c} + \frac{b}{cx}. \]
Now, consider the limit as \( x \rightarrow 0 \):
\[ \lim_{x \rightarrow 0} \frac{b}{cx}. \]
Since \( c \neq 0 \), as \( x \rightarrow 0 \), \[ \frac{b}{cx} \rightarrow \infty \quad \text{(if } b \neq 0\text{)}. \]
Hence, the given limit does not exist for \( b \neq 0 \).
Therefore,
\[ \boxed{ \lim_{x \rightarrow 0} \left( \frac{ax + b}{cx} + 1 \right) \text{ does not exist (diverges), if } b \neq 0. } \]