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