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(5-5*x)/(-1+sqrt(x))
Limit of the function/ (5-5*x)/(-1+sqrt(x))

Limit of the function (5-5*x)/(-1+sqrt(x))

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     / 5 - 5*x  \
 lim |----------|
x->1+|       ___|
     \-1 + \/ x /
lim⁡x→1+(5−5xx−1)\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) 
Limit((5 - 5*x)/(-1 + sqrt(x)), x, 1)
Detail solution
Let's take the limit
lim⁡x→1+(5−5xx−1)\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right)
Multiply numerator and denominator by
−x−1- \sqrt{x} - 1
we get
(5−5x)(−x−1)(−x−1)(x−1)\frac{\left(5 - 5 x\right) \left(- \sqrt{x} - 1\right)}{\left(- \sqrt{x} - 1\right) \left(\sqrt{x} - 1\right)}
=
5(1−x)(−x−1)1−x\frac{5 \left(1 - x\right) \left(- \sqrt{x} - 1\right)}{1 - x}
=
−5x−5- 5 \sqrt{x} - 5
The final answer:
lim⁡x→1+(5−5xx−1)\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right)
=
lim⁡x→1+(−5x−5)\lim_{x \to 1^+}\left(- 5 \sqrt{x} - 5\right)
=
−10-10
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→1+(5−5x)=0\lim_{x \to 1^+}\left(5 - 5 x\right) = 0
and limit for the denominator is
lim⁡x→1+(x−1)=0\lim_{x \to 1^+}\left(\sqrt{x} - 1\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→1+(5−5xx−1)\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right)
=
Let's transform the function under the limit a few
lim⁡x→1+(5(1−x)x−1)\lim_{x \to 1^+}\left(\frac{5 \left(1 - x\right)}{\sqrt{x} - 1}\right)
=
lim⁡x→1+(ddx(5−5x)ddx(x−1))\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(5 - 5 x\right)}{\frac{d}{d x} \left(\sqrt{x} - 1\right)}\right)
=
lim⁡x→1+(−10x)\lim_{x \to 1^+}\left(- 10 \sqrt{x}\right)
=
lim⁡x→1+−10\lim_{x \to 1^+} -10
=
lim⁡x→1+−10\lim_{x \to 1^+} -10
=
−10-10
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
-2.0-1.5-1.0-0.52.00.00.51.01.5-15-5
Plot the graph
Other limits x→0, -oo, +oo, 1
lim⁡x→1−(5−5xx−1)=−10\lim_{x \to 1^-}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = -10
More at x→1 from the left
lim⁡x→1+(5−5xx−1)=−10\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = -10
lim⁡x→∞(5−5xx−1)=−∞\lim_{x \to \infty}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = -\infty
More at x→oo
lim⁡x→0−(5−5xx−1)=−5\lim_{x \to 0^-}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = -5
More at x→0 from the left
lim⁡x→0+(5−5xx−1)=−5\lim_{x \to 0^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = -5
More at x→0 from the right
lim⁡x→−∞(5−5xx−1)=−∞i\lim_{x \to -\infty}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) = - \infty i
More at x→-oo
Rapid solution [src] 
-10
−10-10
Expand and simplify
One‐sided limits [src] 
     / 5 - 5*x  \
 lim |----------|
x->1+|       ___|
     \-1 + \/ x /
lim⁡x→1+(5−5xx−1)\lim_{x \to 1^+}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) 
-10
−10-10 
= -10.0
     / 5 - 5*x  \
 lim |----------|
x->1-|       ___|
     \-1 + \/ x /
lim⁡x→1−(5−5xx−1)\lim_{x \to 1^-}\left(\frac{5 - 5 x}{\sqrt{x} - 1}\right) 
-10
−10-10 
= -10.0
= -10.0
Numerical answer [src] 
-10.0
-10.0
The graph
Limit of the function (5-5*x)/(-1+sqrt(x))
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