Question

In Exercises 5 and \(6 ,\) use \(\lim _ { x \rightarrow c } k = k , \lim _ { x \rightarrow c } x = c ,\) and the properties of limits to find the limit. $$\lim _ { x \rightarrow c } \left( 2 x ^ { 3 } - 3 x ^ { 2 } + x - 1 \right)$$

Short answer

The limit of the function as \(x\) approaches \(c\) is \(2c^3 - 3c^2 + c - 1\).

Step-by-step solution

Break down the function

We can break down the given function into several terms: \(2x^3\), \(-3x^2\), \(x\), and \(-1\) and find the limit of each term separately.

Find the limit of each term

For the first term, \(2x^3\), the limit is \(2c^3\) (apply the limit to the \(x^3\) part and then multiply by the constant 2). For the second term, \(-3x^2\), the limit is \(-3c^2\). For the third term, \(x\), the limit is \(c\). And for the last term, which is a constant, the limit is \(-1\).

Combine the limits

Now we combine these individual limits to find the overall limit of the function. This sum becomes: \(2c^3 - 3c^2 + c - 1\).