Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
- Limit of x^x/factorial(2*x)
-
Limit of x*tan(4*x)/(1-cos(x))
-
Limit of x*(-log(6+x)+log(x))
-
Limit of x/(3+7*x)
-
Identical expressions
- x/(three + seven *x)
- x divide by (3 plus 7 multiply by x)
- x divide by (three plus seven multiply by x)
- x/(3+7x)
- x/3+7x
- x divide by (3+7*x)
-
Similar expressions
- (-2+5*x)/(3+7*x)
- ((2+7*x)/(3+7*x))^(2*x)
- (-2+4*x)/(3+7*x)
- ((-2+7*x)/(3+7*x))^(3-x)
- x/(3-7*x)
Limit of the function x/(3+7*x)
The solution
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/ x \ lim |-------| x->oo\3 + 7*x/
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$
Limit(x/(3 + 7*x), x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{7 + \frac{3}{x}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{7 + \frac{3}{x}} = \lim_{u \to 0^+} \frac{1}{3 u + 7}$$
=
$$\frac{1}{0 \cdot 3 + 7} = \frac{1}{7}$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right) = \frac{1}{7}$$
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$ =
$$\lim_{x \to \infty} \frac{1}{7 + \frac{3}{x}}$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty} \frac{1}{7 + \frac{3}{x}} = \lim_{u \to 0^+} \frac{1}{3 u + 7}$$
=
$$\frac{1}{0 \cdot 3 + 7} = \frac{1}{7}$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right) = \frac{1}{7}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(7 x + 3\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(7 x + 3\right)}\right)$$
=
$$\lim_{x \to \infty} \frac{1}{7}$$
=
$$\lim_{x \to \infty} \frac{1}{7}$$
=
$$\frac{1}{7}$$
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
$$\lim_{x \to \infty} x = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty}\left(7 x + 3\right) = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} x}{\frac{d}{d x} \left(7 x + 3\right)}\right)$$
=
$$\lim_{x \to \infty} \frac{1}{7}$$
=
$$\lim_{x \to \infty} \frac{1}{7}$$
=
$$\frac{1}{7}$$
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
$$\lim_{x \to \infty}\left(\frac{x}{7 x + 3}\right) = \frac{1}{7}$$
$$\lim_{x \to 0^-}\left(\frac{x}{7 x + 3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{7 x + 3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{7 x + 3}\right) = \frac{1}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{7 x + 3}\right) = \frac{1}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{7 x + 3}\right) = \frac{1}{7}$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{7 x + 3}\right) = 0$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{7 x + 3}\right) = 0$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{7 x + 3}\right) = \frac{1}{10}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{7 x + 3}\right) = \frac{1}{10}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{7 x + 3}\right) = \frac{1}{7}$$
More at x→-oo
The graph