Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (7+n)/(5+n)
-
Limit of -2+x^3+6*x
-
Limit of -x+(-2+x)^4/(3+x)^4
-
Limit of (1+x+x^2)/(-2+x^2-x)
- Graphing y =:
- (-1+sqrt(1-x))/x
-
Identical expressions
- (- one +sqrt(one -x))/x
- ( minus 1 plus square root of (1 minus x)) divide by x
- ( minus one plus square root of (one minus x)) divide by x
- (-1+√(1-x))/x
- -1+sqrt1-x/x
- (-1+sqrt(1-x)) divide by x
-
Similar expressions
- (1+sqrt(1-x))/x
- (-1-sqrt(1-x))/x
- (-1+sqrt(1+x))/x
Limit of the function (-1+sqrt(1-x))/x
The solution
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[src]
/ _______\
|-1 + \/ 1 - x |
lim |--------------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
Limit((-1 + sqrt(1 - x))/x, x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
Multiply numerator and denominator by
$$\sqrt{1 - x} + 1$$
we get
$$\frac{\frac{\sqrt{1 - x} - 1}{x} \left(\sqrt{1 - x} + 1\right)}{\sqrt{1 - x} + 1}$$
=
$$- \frac{1}{\sqrt{1 - x} + 1}$$
=
$$- \frac{1}{\sqrt{1 - x} + 1}$$
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{\sqrt{1 - x} + 1}\right)$$
=
$$- \frac{1}{2}$$
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
Multiply numerator and denominator by
$$\sqrt{1 - x} + 1$$
we get
$$\frac{\frac{\sqrt{1 - x} - 1}{x} \left(\sqrt{1 - x} + 1\right)}{\sqrt{1 - x} + 1}$$
=
$$- \frac{1}{\sqrt{1 - x} + 1}$$
=
$$- \frac{1}{\sqrt{1 - x} + 1}$$
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{\sqrt{1 - x} + 1}\right)$$
=
$$- \frac{1}{2}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sqrt{1 - x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sqrt{1 - x} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{2 \sqrt{1 - x}}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+}\left(\sqrt{1 - x} - 1\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} x = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \left(\sqrt{1 - x} - 1\right)}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to 0^+}\left(- \frac{1}{2 \sqrt{1 - x}}\right)$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$\lim_{x \to 0^+} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = - \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = -1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = -1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = -1$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = -1$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sqrt{1 - x} - 1}{x}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/ _______\
|-1 + \/ 1 - x |
lim |--------------|
x->0+\ x /
$$\lim_{x \to 0^+}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
/ _______\
|-1 + \/ 1 - x |
lim |--------------|
x->0-\ x /
$$\lim_{x \to 0^-}\left(\frac{\sqrt{1 - x} - 1}{x}\right)$$
-1/2
$$- \frac{1}{2}$$
= -0.5
= -0.5
The graph