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Limit of the function/ x/(2-x)

Limit of the function x/(2-x)

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      /  x  \
 lim  |-----|
x->-oo\2 - x/

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right)

Limit(x/(2 - x), x, -oo)

Detail solution

Let's take the limit

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right)

Let's divide numerator and denominator by x:

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right)

\lim_{x \to -\infty} \frac{1}{-1 + \frac{2}{x}}

Do Replacement

u = \frac{1}{x}

then

\lim_{x \to -\infty} \frac{1}{-1 + \frac{2}{x}} = \lim_{u \to 0^+} \frac{1}{2 u - 1}

\frac{1}{-1 + 0 \cdot 2} = -1

The final answer:

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right) = -1

Lopital's rule

We have indeterminateness of type

-oo/oo,

i.e. limit for the numerator is

\lim_{x \to -\infty} x = -\infty

and limit for the denominator is

\lim_{x \to -\infty}\left(2 - x\right) = \infty

Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right)

\lim_{x \to -\infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(2 - x\right)}\right)

\lim_{x \to -\infty} -1

\lim_{x \to -\infty} -1

-1

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

Plot the graph

Rapid solution [src]

-1

-1

Expand and simplify

Other limits x→0, -oo, +oo, 1

\lim_{x \to -\infty}\left(\frac{x}{2 - x}\right) = -1

\lim_{x \to \infty}\left(\frac{x}{2 - x}\right) = -1

\lim_{x \to 0^-}\left(\frac{x}{2 - x}\right) = 0

\lim_{x \to 0^+}\left(\frac{x}{2 - x}\right) = 0

\lim_{x \to 1^-}\left(\frac{x}{2 - x}\right) = 1

\lim_{x \to 1^+}\left(\frac{x}{2 - x}\right) = 1

The graph