Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (-exp(-x)+exp(x))/(exp(x)+exp(-x))
-
Limit of x+2*x^3+5*x^4-x^2/3
-
Limit of (-x^3+2*x+5*x^4)/(1+x^4-8*x^3)
-
Limit of ((3+5*x)/(-2+4*x))^(1+3*x)
- Graphing y =:
- (1-x^2)/(1-sqrt(x))
-
Identical expressions
- (one -x^ two)/(one -sqrt(x))
- (1 minus x squared ) divide by (1 minus square root of (x))
- (one minus x to the power of two) divide by (one minus square root of (x))
- (1-x^2)/(1-√(x))
- (1-x2)/(1-sqrt(x))
- 1-x2/1-sqrtx
- (1-x²)/(1-sqrt(x))
- (1-x to the power of 2)/(1-sqrt(x))
- 1-x^2/1-sqrtx
- (1-x^2) divide by (1-sqrt(x))
-
Similar expressions
- (1-x^2)/(1+sqrt(x))
- (1+x^2)/(1-sqrt(x))
Limit of the function (1-x^2)/(1-sqrt(x))
The solution
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[src]
/ 2 \
| 1 - x |
lim |---------|
x->1+| ___|
\1 - \/ x /
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
Limit((1 - x^2)/(1 - sqrt(x)), x, 1)
Detail solution
Let's take the limit
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
Multiply numerator and denominator by
$$\sqrt{x} + 1$$
we get
$$\frac{\left(1 - x^{2}\right) \left(\sqrt{x} + 1\right)}{\left(1 - \sqrt{x}\right) \left(\sqrt{x} + 1\right)}$$
=
$$\frac{\left(-1\right) \left(\sqrt{x} + 1\right) \left(x - 1\right) \left(x + 1\right)}{1 - x}$$
=
$$\left(\sqrt{x} + 1\right) \left(x + 1\right)$$
The final answer:
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
=
$$\lim_{x \to 1^+}\left(\left(\sqrt{x} + 1\right) \left(x + 1\right)\right)$$
=
$$4$$
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
Multiply numerator and denominator by
$$\sqrt{x} + 1$$
we get
$$\frac{\left(1 - x^{2}\right) \left(\sqrt{x} + 1\right)}{\left(1 - \sqrt{x}\right) \left(\sqrt{x} + 1\right)}$$
=
$$\frac{\left(-1\right) \left(\sqrt{x} + 1\right) \left(x - 1\right) \left(x + 1\right)}{1 - x}$$
=
$$\left(\sqrt{x} + 1\right) \left(x + 1\right)$$
The final answer:
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
=
$$\lim_{x \to 1^+}\left(\left(\sqrt{x} + 1\right) \left(x + 1\right)\right)$$
=
$$4$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(1 - x^{2}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(1 - \sqrt{x}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(1 - x^{2}\right)}{\frac{d}{d x} \left(1 - \sqrt{x}\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(4 x^{\frac{3}{2}}\right)$$
=
$$\lim_{x \to 1^+} 4$$
=
$$\lim_{x \to 1^+} 4$$
=
$$4$$
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
$$\lim_{x \to 1^+}\left(1 - x^{2}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(1 - \sqrt{x}\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(1 - x^{2}\right)}{\frac{d}{d x} \left(1 - \sqrt{x}\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(4 x^{\frac{3}{2}}\right)$$
=
$$\lim_{x \to 1^+} 4$$
=
$$\lim_{x \to 1^+} 4$$
=
$$4$$
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
$$\lim_{x \to 1^-}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 4$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = - \infty i$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 4$$
$$\lim_{x \to \infty}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right) = - \infty i$$
More at x→-oo
One‐sided limits
[src]
/ 2 \
| 1 - x |
lim |---------|
x->1+| ___|
\1 - \/ x /
$$\lim_{x \to 1^+}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
4
$$4$$
= 4
/ 2 \
| 1 - x |
lim |---------|
x->1-| ___|
\1 - \/ x /
$$\lim_{x \to 1^-}\left(\frac{1 - x^{2}}{1 - \sqrt{x}}\right)$$
4
$$4$$
= 4
= 4
The graph