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Find the limit of $\frac{x^4-81}{x+3}$ as $x$ approaches $-3$

Step-by-step Solution

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Final answer to the problem

$-108$
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Step-by-step Solution

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  • Solve using L'H么pital's rule
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  • Solve using limit properties
  • Solve using direct substitution
  • Solve the limit using factorization
  • Solve the limit using rationalization
  • Integrate by partial fractions
  • Product of Binomials with Common Term
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If we directly evaluate the limit $\lim_{x\to -3}\left(\frac{x^4-81}{x+3}\right)$ as $x$ tends to $-3$, we can see that it gives us an indeterminate form

$\frac{0}{0}$

Learn how to solve discriminant of quadratic equation problems step by step online.

$\frac{0}{0}$

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Learn how to solve discriminant of quadratic equation problems step by step online. Find the limit of (x^4-81)/(x+3) as x approaches -3. If we directly evaluate the limit \lim_{x\to -3}\left(\frac{x^4-81}{x+3}\right) as x tends to -3, we can see that it gives us an indeterminate form. We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately. After deriving both the numerator and denominator, the limit results in. The limit of the product of a function and a constant is equal to the limit of the function, times the constant: \displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}.

Final answer to the problem

$-108$

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Function Plot

Plotting: $\frac{x^4-81}{x+3}$

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5
6
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

How to improve your answer:

Main Topic: Discriminant of Quadratic Equation

Quadratic equations are those algebraic equations of the form ax^2+bx+c, where a, b, and c are constant values. The discriminant of a quadratic equation is calculated using the formula D=b^2-4ac, and it helps us to determine how many roots an equation of this type has. When D>0 the equation has two real roots, when D<0 the equation has no real roots, and when D=0 the equation has a repeated real root.

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