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Limits by factoring Calculator

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1

Solved example of limits by factoring

$\frac{\lim_{x\to-1}\left(x^4-2x+3\right)}{x^4-2x-3}$
2

We can factor the polynomial $x^4-2x-3$ using synthetic division (Ruffini's rule). We search for a root in the factors of the constant term $-3$ and we found that $-1$ is a root of the polynomial

${\left(-1\right)}^4-2-1-3=0$
3

Let's divide the polynomial by $x+1$ using synthetic division. First, write the coefficients of the terms of the numerator in descending order. Then, take the first coefficient $1$ and multiply by the factor $-1$. Add the result to the second coefficient and then multiply this by $-1$ and so on

$\left|\begin{array}{c}1 & 0 & 0 & -2 & -3 \\ & -1 & 1 & -1 & 3 \\ 1 & -1 & 1 & -3 & 0\end{array}\right|-1$
4

In the last row of the division appear the new coefficients, with remainder equals zero. Now, rewrite the polynomial (a degree less) with the new coefficients, and multiplied by the factor $x+1$

$\frac{\lim_{x\to-1}\left(x^4-2x+3\right)}{\left(x^{3}-x^{2}+x-3\right)\left(x+1\right)}$
5

Factor by $x^{2}$

$\frac{\lim_{x\to-1}\left(x^4-2x+3\right)}{\left(x^{2}\left(-1+x^{\left(3-2\right)}\right)+x-3\right)\left(x+1\right)}$
6

Subtract the values $3$ and $-2$

$\frac{\lim_{x\to-1}\left(x^4-2x+3\right)}{\left(x^{2}\left(-1+x^{1}\right)+x-3\right)\left(x+1\right)}$
7

Any expression to the power of $1$ is equal to that same expression

$\frac{\lim_{x\to-1}\left(x^4-2x+3\right)}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\frac{\lim_{x\to-1}\left(x^4\right)+\lim_{x\to-1}\left(-2x+3\right)}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\frac{\lim_{x\to-1}\left(x^4\right)+\lim_{x\to-1}\left(-2x\right)+\lim_{x\to-1}\left(3\right)}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
8

The limit of a sum of two functions is equal to the sum of the limits of each function: $\displaystyle\lim_{x\to c}(f(x)\pm g(x))=\lim_{x\to c}(f(x))\pm\lim_{x\to c}(g(x))$

$\frac{\lim_{x\to-1}\left(x^4\right)+\lim_{x\to-1}\left(-2x\right)+\lim_{x\to-1}\left(3\right)}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
9

The limit of a constant is just the constant

$\frac{\lim_{x\to-1}\left(x^4\right)+\lim_{x\to-1}\left(-2x\right)+3}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
10

Evaluating the limit when $x$ tends to $-1$

$\frac{{\left(-1\right)}^4+\lim_{x\to-1}\left(-2x\right)+3}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$

Calculate the power ${\left(-1\right)}^4$

$1$
11

Simplifying

$\frac{1+\lim_{x\to-1}\left(-2x\right)+3}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
12

Add the values $1$ and $3$

$\frac{4+\lim_{x\to-1}\left(-2x\right)}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
13

Evaluating the limit when $x$ tends to $-1$

$\frac{4-2-1}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$

Multiply $-2$ times $-1$

$2$
14

Simplifying

$\frac{4+2}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$
15

Add the values $4$ and $2$

$\frac{6}{\left(x^{2}\left(-1+x\right)+x-3\right)\left(x+1\right)}$

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