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Find the limit of $\frac{x^2+8x-9}{x^3-x^2+5x-5}$ as $x$ approaches $5$

Step-by-step Solution

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Final answer to the problem

$\frac{7}{15}$
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Step-by-step Solution

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  • Solve the limit using factorization
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We can factor the polynomial $x^3-x^2+5x-5$ 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 $-5$

$1, 5$

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$1, 5$

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Learn how to solve limits by direct substitution problems step by step online. Find the limit of (x^2+8x+-9)/(x^3-x^25x+-5) as x approaches 5. We can factor the polynomial x^3-x^2+5x-5 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 -5. Next, list all divisors of the leading coefficient a_n, which equals 1. The possible roots \pm\frac{p}{q} of the polynomial x^3-x^2+5x-5 will then be. Trying all possible roots, we found that 1 is a root of the polynomial. When we evaluate it in the polynomial, it gives us 0 as a result.

Final answer to the problem

$\frac{7}{15}$

Exact Numeric Answer

$0.4667$

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Function Plot

Plotting: $\frac{x^2+8x-9}{x^3-x^2+5x-5}$

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7
8
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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Main Topic: Limits by Direct Substitution

Find limits of functions at a specific point by directly plugging the value into the function.

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