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(5+3*x)/(7+2*x)
Limit of the function/ (5+3*x)/(7+2*x)

Limit of the function (5+3*x)/(7+2*x)

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     /5 + 3*x\
 lim |-------|
x->oo\7 + 2*x/
lim⁡x→∞(3x+52x+7)\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right) 
Limit((5 + 3*x)/(7 + 2*x), x, oo, dir='-')
Detail solution
Let's take the limit
lim⁡x→∞(3x+52x+7)\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right)
Let's divide numerator and denominator by x:
lim⁡x→∞(3x+52x+7)\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right) =
lim⁡x→∞(3+5x2+7x)\lim_{x \to \infty}\left(\frac{3 + \frac{5}{x}}{2 + \frac{7}{x}}\right)
Do Replacement
u=1xu = \frac{1}{x}
then
lim⁡x→∞(3+5x2+7x)=lim⁡u→0+(5u+37u+2)\lim_{x \to \infty}\left(\frac{3 + \frac{5}{x}}{2 + \frac{7}{x}}\right) = \lim_{u \to 0^+}\left(\frac{5 u + 3}{7 u + 2}\right)
=
0⋅5+30⋅7+2=32\frac{0 \cdot 5 + 3}{0 \cdot 7 + 2} = \frac{3}{2}

The final answer:
lim⁡x→∞(3x+52x+7)=32\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{3}{2}
Lopital's rule
We have indeterminateness of type
oo/oo,

i.e. limit for the numerator is
lim⁡x→∞(3x+5)=∞\lim_{x \to \infty}\left(3 x + 5\right) = \infty
and limit for the denominator is
lim⁡x→∞(2x+7)=∞\lim_{x \to \infty}\left(2 x + 7\right) = \infty
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→∞(3x+52x+7)\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right)
=
lim⁡x→∞(ddx(3x+5)ddx(2x+7))\lim_{x \to \infty}\left(\frac{\frac{d}{d x} \left(3 x + 5\right)}{\frac{d}{d x} \left(2 x + 7\right)}\right)
=
lim⁡x→∞32\lim_{x \to \infty} \frac{3}{2}
=
lim⁡x→∞32\lim_{x \to \infty} \frac{3}{2}
=
32\frac{3}{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
02468-8-6-4-2-1010-100100
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Rapid solution [src] 
3/2
32\frac{3}{2}
Expand and simplify
Other limits x→0, -oo, +oo, 1
lim⁡x→∞(3x+52x+7)=32\lim_{x \to \infty}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{3}{2}
lim⁡x→0−(3x+52x+7)=57\lim_{x \to 0^-}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{5}{7}
More at x→0 from the left
lim⁡x→0+(3x+52x+7)=57\lim_{x \to 0^+}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{5}{7}
More at x→0 from the right
lim⁡x→1−(3x+52x+7)=89\lim_{x \to 1^-}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{8}{9}
More at x→1 from the left
lim⁡x→1+(3x+52x+7)=89\lim_{x \to 1^+}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{8}{9}
More at x→1 from the right
lim⁡x→−∞(3x+52x+7)=32\lim_{x \to -\infty}\left(\frac{3 x + 5}{2 x + 7}\right) = \frac{3}{2}
More at x→-oo
The graph
Limit of the function (5+3*x)/(7+2*x)
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