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For the following exercise, evaluate the limits algebraically.

limx2(x25x+6x+2)\lim _{x \rightarrow 2}\left(\frac{x^2-5 x+6}{x+2}\right)


Answered 2 years ago
Answered 2 years ago
Step 1
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To find the limit limx2(x25x+6x+2)\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

limxaf(x)g(x)=limxaf(x)limxag(x)=AB,B0(1)\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

limxa[kf(x)]=klimxaf(x)=kA(2)\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

limxa[f(x)g(x)]=limxaf(x)limxag(x)=AB(3)\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

limxa[f(x)]n=[limxf(x)]n=An,    n is positive integer(4)\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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