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# Find the limit of $\frac{x^2-25}{x-5}$ as $x$ approaches $5$

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##  Final answer to the problem

$10$
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##  Step-by-step Solution 

Specify the solving method

1

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

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

Calculate the power $\sqrt{25}$

$\lim_{x\to5}\left(\frac{\left(x+5\right)\left(\sqrt{x^2}-\sqrt{25}\right)}{x-5}\right)$
3

Simplify $\sqrt{x^2}$ using the power of a power property: $\left(a^m\right)^n=a^{m\cdot n}$. In the expression, $m$ equals $2$ and $n$ equals $\frac{1}{2}$

$\lim_{x\to5}\left(\frac{\left(x+5\right)\left(x-\sqrt{25}\right)}{x-5}\right)$
4

Calculate the power $\sqrt{25}$

$\lim_{x\to5}\left(\frac{\left(x+5\right)\left(x- 5\right)}{x-5}\right)$
5

Multiply $-1$ times $5$

$\lim_{x\to5}\left(\frac{\left(x+5\right)\left(x-5\right)}{x-5}\right)$
6

Simplify the fraction $\frac{\left(x+5\right)\left(x-5\right)}{x-5}$ by $x-5$

$\lim_{x\to5}\left(x+5\right)$
7

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

$5+5$
8

Add the values $5$ and $5$

$10$

##  Final answer to the problem

$10$

##  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

Solve using direct substitutionSolve using limit propertiesSolve using L'HÃ´pital's ruleSolve using rationalizationSolve without using l'HÃ´pital

SnapXam A2

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4
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

###  Main Topic: Limits by Direct Substitution

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