$\lim_{x\to1}\left(\frac{x^5-1}{x^3-1}\right)$

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Final answer to the problem

$\frac{5}{3}$
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Step-by-step Solution

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Apply the formula: $a^3+b$$=\left(a-\sqrt[3]{\left|b\right|}\right)\left(a^2+a\sqrt[3]{\left|b\right|}+\sqrt[3]{\left|b\right|^{2}}\right)$, where $a=x$ and $b=-1$

$\lim_{x\to1}\left(\frac{x^5-1}{\left(x-1\right)\left(x^2+x+1\right)}\right)$

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$\lim_{x\to1}\left(\frac{x^5-1}{\left(x-1\right)\left(x^2+x+1\right)}\right)$

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Learn how to solve problems step by step online. (x)->(1)lim((x^5-1)/(x^3-1)). Apply the formula: a^3+b=\left(a-\sqrt[3]{\left|b\right|}\right)\left(a^2+a\sqrt[3]{\left|b\right|}+\sqrt[3]{\left|b\right|^{2}}\right), where a=x and b=-1. We can factor the polynomial x^5-1 using the rational root theorem, which guarantees that for a polynomial of the form a_nx^n+a_{n-1}x^{n-1}+\dots+a_0 there is a rational root of the form \pm\frac{p}{q}, where p belongs to the divisors of the constant term a_0, and q belongs to the divisors of the leading coefficient a_n. List all divisors p of the constant term a_0, which equals -1. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^5-1 will then be.

Final answer to the problem

$\frac{5}{3}$

Exact Numeric Answer

$1.6666667$

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Function Plot

Plotting: $\frac{x^5-1}{x^3-1}$

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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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