Result of Rule Application

Rule Applied

$\underset{x\to 3}{lim}\left(2x-4\right)$

$\=\underset{x\to 3}{lim}\left(2x\right)-\underset{x\to 3}{lim}4$

difference rule

$\=\underset{x\to 3}{lim}\left(2x\right)-4$

constant rule

$\=2 \left(\underset{x\to 3}{lim}x\right)-4$

constant multiple rule

$\=2\left(3\right)-4$

identity rule

$\=2$

basic arithmetic ⇒ final answer

Table 1.3.1(a)   Stepwise evaluation of $\underset{x\to 3}{lim}\left(2x-4\right)\=2$ 

Annotated stepwise solution via the Context Panel

Loading Student:-Calculus1

$\underset{x\to 3}{lim}\left(2 x-4\right)$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Limit Steps}\\ & & \underset{x\to 3}{lim}\left(2x-4\right)\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{sum}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{sum}}\text{rule}\\ & & \underset{x\to a}{lim}\left(f\left(x\right)+g\left(x\right)\right)=\left(\underset{x\to a}{lim}f\left(x\right)\right)+\left(\underset{x\to a}{lim}g\left(x\right)\right)\\ & & f\left(x\right)=2x\\ & & g\left(x\right)=\mathrm{-4}\\ & & \text{This gives:}\\ & & \left(\underset{x\to 3}{lim}2x\right)+\underset{x\to 3}{lim}\left(\mathrm{-4}\right)\\ \text{▫}& & \text{2. Apply the}\mathbf{\text{constant}}\text{rule to the term}\underset{x\to 3}{lim}\left(\mathrm{-4}\right)\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant}}\text{rule}\\ & & \mathrm{Limit}\left(C\,x\right)=C\\ & \text{◦}& \text{This means}\\ & & \underset{x\to 3}{lim}\left(\mathrm{-4}\right)=\mathrm{-4}\\ & & \text{We can now rewrite the limit as:}\\ & & \left(\underset{x\to 3}{lim}2x\right)-4\\ \text{▫}& & \text{3. Apply the}\mathbf{\text{constant multiple}}\text{rule to the term}\underset{x\to 3}{lim}2x\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{constant multiple}}\text{rule}\\ & & \underset{x\to 3}{lim}Cf\left(x\right)=C\left(\underset{x\to 3}{lim}f\left(x\right)\right)\\ & \text{◦}& \text{This means:}\\ & & \underset{x\to 3}{lim}2x=2\cdot \left(\underset{x\to 3}{lim}x\right)\\ & & \text{We can rewrite the limit as:}\\ & & 2\left(\underset{x\to 3}{lim}x\right)-4\\ \text{▫}& & \text{4. Apply the}\mathbf{\text{identity}}\text{rule}\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{identity}}\text{rule}\\ & & \underset{x\to a}{lim}x=a\\ & & \text{This gives:}\\ & & 2\end{array}$

Table 1.3.1(b)   Maple's stepwise solution via the All Solution Steps option in the Context Panel