# Find the limit $\lim_{x\to5}\left(\frac{x^2-25}{x-5}\right)$

## Step-by-step Solution

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$10$
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##  Step-by-step Solution 

Problem to solve:

$\lim_{x\to5}\left(\frac{x^2-25}{x-5}\right)$

Specify the solving method

Plug in the value $5$ into the limit

$\frac{5^2-25}{5-5}$

Calculate the power $5^2$

$\frac{25-25}{5-5}$

Subtract the values $25$ and $-25$

$\frac{0}{5-5}$

Subtract the values $5$ and $-5$

$\frac{0}{0}$
1

If we directly evaluate the limit $\lim_{x\to 5}\left(\frac{x^2-25}{x-5}\right)$ as $x$ tends to $5$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
2

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

$\lim_{x\to 5}\left(\frac{\frac{d}{dx}\left(x^2-25\right)}{\frac{d}{dx}\left(x-5\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(x^2-25\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-25\right)$

The derivative of the constant function ($-25$) is equal to zero

$\frac{d}{dx}\left(x^2\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$2x$

Find the derivative of the denominator

$\frac{d}{dx}\left(x-5\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-5\right)$

The derivative of the constant function ($-5$) is equal to zero

$\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$1$

Any expression divided by one ($1$) is equal to that same expression

$\lim_{x\to5}\left(2x\right)$
3

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to5}\left(2x\right)$
4

The limit of the product of a function and a constant is equal to the limit of the function, times the constant: $\displaystyle \lim_{t\to 0}{\left(at\right)}=a\cdot\lim_{t\to 0}{\left(t\right)}$

$2\lim_{x\to5}\left(x\right)$
5

Evaluate the limit $\lim_{x\to5}\left(x\right)$ by replacing all occurrences of $x$ by $5$

$2\cdot 5$
6

Multiply $2$ times $5$

$10$

$10$

$10$

##  Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Limits by Direct SubstitutionLimits by factoringLimits by rationalizing

SnapXam A2

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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

### Main topic:

Limits by Direct Substitution

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