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Question

# For the following exercise, evaluate the limits algebraically.$\lim _{x \rightarrow 2}\left(\frac{x^2-5 x+6}{x+2}\right)$

Solution

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To find the limit $\lim_{x\to 2}\left(\dfrac{x^2-5x+6}{x+2}\right)$, we can use the following properties

We will use the quotient of functions property which states

\begin{aligned} \lim_{x\to a}\frac{f(x)}{g(x)}=\frac{\lim_{x\to a}f(x)}{\lim_{x\to a}g(x)}=\frac{A}{B},\quad B\not=0\tag{1}\end{aligned}

We will use constant times a function property which states

\begin{aligned}\lim_{x\to a}[k\cdot f(x)]=k\lim_{x\to a}f(x)=kA\tag{2}\end{aligned}

We will use difference of functions property which states

\begin{aligned}\lim_{x\to a}[f(x)-g(x)]=\lim_{x\to a}f(x)-\lim_{x\to a}g(x)=A-B\tag{3}\end{aligned}

We will use function raised to an exponent property which states

\begin{aligned}\lim_{x\to a}[ f(x)]^n=\left[\lim_{x\to \infty} f(x)\right]^n=A^n,\;\;\text{n is positive integer}\tag{4}\end{aligned}

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