Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of ((-2+3*x)/(1+3*x))^(2*x)
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Limit of ((-4+3*x)/(2+3*x))^(2*x)
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Limit of (1-cos(4*x))/(1-cos(8*x))
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Limit of (1+x^2-4*x)/(1+2*x)
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Identical expressions
- (one +x^ two - four *x)/(one + two *x)
- (1 plus x squared minus 4 multiply by x) divide by (1 plus 2 multiply by x)
- (one plus x to the power of two minus four multiply by x) divide by (one plus two multiply by x)
- (1+x2-4*x)/(1+2*x)
- 1+x2-4*x/1+2*x
- (1+x²-4*x)/(1+2*x)
- (1+x to the power of 2-4*x)/(1+2*x)
- (1+x^2-4x)/(1+2x)
- (1+x2-4x)/(1+2x)
- 1+x2-4x/1+2x
- 1+x^2-4x/1+2x
- (1+x^2-4*x) divide by (1+2*x)
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Similar expressions
- (1+x^2-4*x)/(1-2*x)
- (1-x^2-4*x)/(1+2*x)
- (1+x^2+4*x)/(1+2*x)
Limit of the function (1+x^2-4*x)/(1+2*x)
The solution
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[src]
/ 2 \
|1 + x - 4*x|
lim |------------|
x->2+\ 1 + 2*x /
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
Limit((1 + x^2 - 4*x)/(1 + 2*x), x, 2)
Detail solution
Let's take the limit
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4 x + 1}{2 x + 1}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4 x + 1}{2 x + 1}\right) = $$
$$\frac{- 8 + 1 + 2^{2}}{1 + 2 \cdot 2} = $$
The final answer:
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{3}{5}$$
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
transform
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4 x + 1}{2 x + 1}\right)$$
=
$$\lim_{x \to 2^+}\left(\frac{x^{2} - 4 x + 1}{2 x + 1}\right) = $$
$$\frac{- 8 + 1 + 2^{2}}{1 + 2 \cdot 2} = $$
= -3/5
The final answer:
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{3}{5}$$
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 2^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{3}{5}$$
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{3}{5}$$
$$\lim_{x \to \infty}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{2}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{2}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = -\infty$$
More at x→-oo
More at x→2 from the left
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{3}{5}$$
$$\lim_{x \to \infty}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{2}{3}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = - \frac{2}{3}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right) = -\infty$$
More at x→-oo
One‐sided limits
[src]
/ 2 \
|1 + x - 4*x|
lim |------------|
x->2+\ 1 + 2*x /
$$\lim_{x \to 2^+}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
-3/5
$$- \frac{3}{5}$$
= -0.6
/ 2 \
|1 + x - 4*x|
lim |------------|
x->2-\ 1 + 2*x /
$$\lim_{x \to 2^-}\left(\frac{- 4 x + \left(x^{2} + 1\right)}{2 x + 1}\right)$$
-3/5
$$- \frac{3}{5}$$
= -0.6
= -0.6
The graph