Step 1:

First, we will try and substitute the limit into the function. So, substituting the 0 into the function, we have:

[latex]\large{\lim_{x \to 0}\frac{3x^2 – 2x}{x} = \frac{3(0)^2 – 2(0)}{0} = \frac{0}{0}}[/latex]

So, we end up with [latex]\frac{0}{0}[/latex] which is an indeterminate form.

Step 2:

Since we ended up with an indeterminate form, we need to rewrite the original function in another way, and then try to solve the limit again.

Notice how we can factor out the x from the numerator, as follows:

[latex]\large{\lim_{x\to 0}\frac{3x^2-2x}{x} = \lim_{x \to 0}(\frac{x(3x-2)}{x}) =\lim_{x \to 0} (3x – 2)}[/latex]

As you see, with some algebraic manipulation, we were able to rewrite the original function. 

Step 3:

Now we can simply plug in the limit and solve, as follows:

[latex]\large{\lim_{x\to 0}3x – 2 = 3(0) – 2 = -2}[/latex]

Solution Complete!