Mister Exam
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How to use it?
- Limit of the function:
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Limit of (2+x^2+3*x)/(2+2*x^2+5*x)
-
Limit of (-51+x^2-14*x)/(-21+x^2-4*x)
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Limit of (-1+x)/(3-2*x)
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Limit of -sec(x)+tan(x)
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Identical expressions
- (- one +x)/(three - two *x)
- ( minus 1 plus x) divide by (3 minus 2 multiply by x)
- ( minus one plus x) divide by (three minus two multiply by x)
- (-1+x)/(3-2x)
- -1+x/3-2x
- (-1+x) divide by (3-2*x)
-
Similar expressions
- (-1+x)/(3+2*x)
- (1+x)/(3-2*x)
- (-1-x)/(3-2*x)
Limit of the function (-1+x)/(3-2*x)
The solution
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/ -1 + x\ lim |-------| x->oo\3 - 2*x/
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$
Limit((-1 + x)/(3 - 2*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{1}{x}}{-2 + \frac{3}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{1}{x}}{-2 + \frac{3}{x}}\right) = \lim_{u \to 0^+}\left(\frac{1 - u}{3 u - 2}\right)$$
=
$$\frac{1 - 0}{-2 + 0 \cdot 3} = - \frac{1}{2}$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$ =
$$\lim_{x \to \infty}\left(\frac{1 - \frac{1}{x}}{-2 + \frac{3}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{1 - \frac{1}{x}}{-2 + \frac{3}{x}}\right) = \lim_{u \to 0^+}\left(\frac{1 - u}{3 u - 2}\right)$$
=
$$\frac{1 - 0}{-2 + 0 \cdot 3} = - \frac{1}{2}$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{2}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x - 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(3 - 2 x\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \left(3 - 2 x\right)}\right)$$
=
$$\lim_{x \to \infty} - \frac{1}{2}$$
=
$$\lim_{x \to \infty} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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)
oo/-oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty}\left(x - 1\right) = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(3 - 2 x\right) = -\infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(x - 1\right)}{\frac{d}{d x} \left(3 - 2 x\right)}\right)$$
=
$$\lim_{x \to \infty} - \frac{1}{2}$$
=
$$\lim_{x \to \infty} - \frac{1}{2}$$
=
$$- \frac{1}{2}$$
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
$$\lim_{x \to \infty}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{2}$$
$$\lim_{x \to 0^-}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{3}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 1}{3 - 2 x}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{3 - 2 x}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{2}$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{3}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{3}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x - 1}{3 - 2 x}\right) = 0$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x - 1}{3 - 2 x}\right) = 0$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x - 1}{3 - 2 x}\right) = - \frac{1}{2}$$
More at x→-oo
The graph