Q. 27 - Calculus

Question

Calculate the limits using only the continuity of linear and power functions and the limit rules. Cite each limit rule that you apply.

$\underset{x\to 3}{\lim }\left(3x+x^2\left(2x+1\right)\right)$

Step-by-Step Solution

Answer

On calculating the limits, we get,

$72$

Step 1. Given information.

Consider the given question,

$\underset{x\to 3}{\lim }\left(3x+x^2\left(2x+1\right)\right)$

Step 2. Calculate the limits.

Using the sum rule,

$=\underset{x\to 3}{\lim }\left(3x\right)+\underset{x\to 3}{\lim }\left(x^2\left(2x+1\right)\right)$

Using the product rule,

$$\begin{gathered} =\underset{x\to 3}{\lim }\left(3x\right)+\underset{x\to 3}{\lim }\left(x^2\right)\underset{x\to 3}{\lim }\left(2x+1\right) \\ =3\left(3\right)+3^2\left(2·3+1\right) \\ =9+9\left(6+1\right) \\ =9+9\left(7\right) \\ =72 \end{gathered}$$