To solve the limit of the given expression:

\[
\lim_{{x \to 0}} \frac{4x^2 - 5x}{x}
\]

### Step 1: Simplify the expression

We can simplify the expression by canceling out the common \(x\) in both the numerator and denominator. The numerator is:

\[
4x^2 - 5x = x(4x - 5)
\]

Thus, the entire expression becomes:

\[
\frac{x(4x - 5)}{x}
\]

Since \(x \neq 0\) (otherwise division by zero occurs), we can cancel the \(x\) terms:

\[
4x - 5
\]

### Step 2: Apply the limit

Now, apply the limit as \(x \to 0\):

\[
\lim_{{x \to 0}} (4x - 5) = 4(0) - 5 = -5
\]

### Final Answer:
\[
\lim_{{x \to 0}} \frac{4x^2 - 5x}{x} = -5
\]

Would you like more details or have any questions about this solution?

Here are 5 related questions for practice:
1. What is \(\lim_{{x \to 0}} \frac{2x^3 - 7x}{x}\)?
2. Find \(\lim_{{x \to 1}} \frac{3x^2 - 2x}{x-1}\).
3. Evaluate \(\lim_{{x \to 0}} \frac{x^2 - x}{x}\).
4. Determine \(\lim_{{x \to 2}} \frac{x^2 - 4x + 4}{x-2}\).
5. Simplify and find \(\lim_{{x \to 0}} \frac{x^2 - 6x}{x^2}\).

**Tip:** When simplifying limits, always check for factors that can cancel out, especially when the function has a potential for indeterminate forms like \(\frac{0}{0}\).

Learn how to solve the limit of the expression (4x^2 - 5x) / x as x approaches 0 with a clear step-by-step explanation. This problem covers important calculus concepts like algebraic simplification and limit evaluation. Ideal for students in Grades 10-12.

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