Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of (1+2*n)/(-1+3*n)
-
Limit of (x^3-3^x)/(-3+x)
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Limit of -x+(-2+x)^4/(3+x)^4
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Identical expressions
- sin(seven *x)/(six *x)
- sinus of (7 multiply by x) divide by (6 multiply by x)
- sinus of (seven multiply by x) divide by (six multiply by x)
- sin(7x)/(6x)
- sin7x/6x
- sin(7*x) divide by (6*x)
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Similar expressions
- (sin(4*x)+sin(7*x))/(6*x)
Limit of the function sin(7*x)/(6*x)
The solution
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[src]
/sin(7*x)\ lim |--------| x->0+\ 6*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
Limit(sin(7*x)/((6*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
Do replacement
$$u = 7 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \lim_{u \to 0^+}\left(\frac{7 \sin{\left(u \right)}}{6 u}\right)$$
=
$$\frac{7 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{6}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{7}{6}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
Do replacement
$$u = 7 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \lim_{u \to 0^+}\left(\frac{7 \sin{\left(u \right)}}{6 u}\right)$$
=
$$\frac{7 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{6}$$
The limit
$$\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)$$
is first remarkable limit, is equal to 1.
The final answer:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{7}{6}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(7 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(7 x \right)}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cos{\left(7 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+} \frac{7}{6}$$
=
$$\lim_{x \to 0^+} \frac{7}{6}$$
=
$$\frac{7}{6}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(7 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(6 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(7 x \right)}}{\frac{d}{d x} 6 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{7 \cos{\left(7 x \right)}}{6}\right)$$
=
$$\lim_{x \to 0^+} \frac{7}{6}$$
=
$$\lim_{x \to 0^+} \frac{7}{6}$$
=
$$\frac{7}{6}$$
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
One‐sided limits
[src]
/sin(7*x)\ lim |--------| x->0+\ 6*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
7/6
$$\frac{7}{6}$$
= 1.16666666666667
/sin(7*x)\ lim |--------| x->0-\ 6*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right)$$
7/6
$$\frac{7}{6}$$
= 1.16666666666667
= 1.16666666666667
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{7}{6}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{7}{6}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{\sin{\left(7 \right)}}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{\sin{\left(7 \right)}}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{7}{6}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{\sin{\left(7 \right)}}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = \frac{\sin{\left(7 \right)}}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(7 x \right)}}{6 x}\right) = 0$$
More at x→-oo
The graph