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# Find the limit $\lim_{x\to10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$

## Step-by-step Solution

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###  Videos

$\frac{1}{8}$
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##  Step-by-step Solution 

Problem to solve:

$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$

Specify the solving method

1

If we directly evaluate the limit $\lim_{x\to 10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$ as $x$ tends to $10$, we can see that it gives us an indeterminate form

$\frac{0}{0}$
2

We can solve this limit by applying L'Hôpital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to 10}\left(\frac{\frac{d}{dx}\left(\sqrt{x+6}-4\right)}{\frac{d}{dx}\left(x-10\right)}\right)$

Learn how to solve limits by direct substitution problems step by step online.

$\frac{0}{0}$

Learn how to solve limits by direct substitution problems step by step online. Find the limit (x)->(10)lim(((x+6)^1/2-4)/(x-10)). If we directly evaluate the limit \lim_{x\to 10}\left(\frac{\sqrt{x+6}-4}{x-10}\right) as x tends to 10, we can see that it gives us an indeterminate form. We can solve this limit by applying L'Hôpital'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)}.

$\frac{1}{8}$

$0.125$

##  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

### beta Got a different answer? Verify it!

Go!
1
2
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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

$\lim_{x\to\:10}\left(\frac{\sqrt{x+6}-4}{x-10}\right)$