Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
-
Limit of (x/2)^(1/(-2+x))
-
Limit of (1+e^x)^(1/x)
-
Limit of (2-7*x+3*x^2)/(2-5*x+2*x^2)
-
Limit of (e^x-e^2)/(-2+x)
- Graphing y =:
- (2-7*x+3*x^2)/(2-5*x+2*x^2)
-
Identical expressions
- (two - seven *x+ three *x^ two)/(two - five *x+ two *x^ two)
- (2 minus 7 multiply by x plus 3 multiply by x squared ) divide by (2 minus 5 multiply by x plus 2 multiply by x squared )
- (two minus seven multiply by x plus three multiply by x to the power of two) divide by (two minus five multiply by x plus two multiply by x to the power of two)
- (2-7*x+3*x2)/(2-5*x+2*x2)
- 2-7*x+3*x2/2-5*x+2*x2
- (2-7*x+3*x²)/(2-5*x+2*x²)
- (2-7*x+3*x to the power of 2)/(2-5*x+2*x to the power of 2)
- (2-7x+3x^2)/(2-5x+2x^2)
- (2-7x+3x2)/(2-5x+2x2)
- 2-7x+3x2/2-5x+2x2
- 2-7x+3x^2/2-5x+2x^2
- (2-7*x+3*x^2) divide by (2-5*x+2*x^2)
-
Similar expressions
- (2-7*x+3*x^2)/(2-5*x-2*x^2)
- (2+7*x+3*x^2)/(2-5*x+2*x^2)
- (2-7*x+3*x^2)/(2+5*x+2*x^2)
- (2-7*x-3*x^2)/(2-5*x+2*x^2)
Limit of the function (2-7*x+3*x^2)/(2-5*x+2*x^2)
The solution
You have entered
[src]
/ 2\
|2 - 7*x + 3*x |
lim |--------------|
x->2+| 2|
\2 - 5*x + 2*x /
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
Limit((2 - 7*x + 3*x^2)/(2 - 5*x + 2*x^2), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\left(x - 2\right) \left(3 x - 1\right)}{\left(x - 2\right) \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{3 x - 1}{2 x - 1}\right) = $$
$$\frac{-1 + 2 \cdot 3}{-1 + 2 \cdot 2} = $$
The final answer:
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{5}{3}$$
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\left(x - 2\right) \left(3 x - 1\right)}{\left(x - 2\right) \left(2 x - 1\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{3 x - 1}{2 x - 1}\right) = $$
$$\frac{-1 + 2 \cdot 3}{-1 + 2 \cdot 2} = $$
= 5/3
The final answer:
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{5}{3}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 2^+}\left(3 x^{2} - 7 x + 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(2 x^{2} - 5 x + 2\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} - 7 x + 2}{2 x^{2} - 5 x + 2}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} - 7 x + 2\right)}{\frac{d}{d x} \left(2 x^{2} - 5 x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{6 x - 7}{4 x - 5}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{6 x - 7}{4 x - 5}\right)$$
=
$$\frac{5}{3}$$
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
$$\lim_{x \to 2^+}\left(3 x^{2} - 7 x + 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 2^+}\left(2 x^{2} - 5 x + 2\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} - 7 x + 2}{2 x^{2} - 5 x + 2}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{\frac{d}{d x} \left(3 x^{2} - 7 x + 2\right)}{\frac{d}{d x} \left(2 x^{2} - 5 x + 2\right)}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{6 x - 7}{4 x - 5}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{6 x - 7}{4 x - 5}\right)$$
=
$$\frac{5}{3}$$
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
$$\lim_{x \to 2^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{5}{3}$$
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{3}{2}$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{3}{2}$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{5}{3}$$
$$\lim_{x \to \infty}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{3}{2}$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 2$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = 2$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right) = \frac{3}{2}$$
More at x→-oo
One‐sided limits
[src]
/ 2\
|2 - 7*x + 3*x |
lim |--------------|
x->2+| 2|
\2 - 5*x + 2*x /
$$\lim_{x \to 2^+}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
/ 2\
|2 - 7*x + 3*x |
lim |--------------|
x->2-| 2|
\2 - 5*x + 2*x /
$$\lim_{x \to 2^-}\left(\frac{3 x^{2} + \left(2 - 7 x\right)}{2 x^{2} + \left(2 - 5 x\right)}\right)$$
5/3
$$\frac{5}{3}$$
= 1.66666666666667
= 1.66666666666667
The graph