Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
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Limit of 10+x^2+3*x^3+8*x
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Limit of (-2+sqrt(x))/(-4+x^2)
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Limit of x+2*x^3+5*x^4-x^2/3
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Limit of (-x^3+2*x+5*x^4)/(1+x^4-8*x^3)
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Identical expressions
- sin(nine *x)/(eight *x)
- sinus of (9 multiply by x) divide by (8 multiply by x)
- sinus of (nine multiply by x) divide by (eight multiply by x)
- sin(9x)/(8x)
- sin9x/8x
- sin(9*x) divide by (8*x)
Limit of the function sin(9*x)/(8*x)
The solution
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[src]
/sin(9*x)\ lim |--------| x->0+\ 8*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
Limit(sin(9*x)/((8*x)), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
Do replacement
$$u = 9 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \lim_{u \to 0^+}\left(\frac{9 \sin{\left(u \right)}}{8 u}\right)$$
=
$$\frac{9 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{8}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{9}{8}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
Do replacement
$$u = 9 x$$
then
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \lim_{u \to 0^+}\left(\frac{9 \sin{\left(u \right)}}{8 u}\right)$$
=
$$\frac{9 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{8}$$
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:
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{9}{8}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(9 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(8 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(9 x \right)}}{\frac{d}{d x} 8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{9 \cos{\left(9 x \right)}}{8}\right)$$
=
$$\lim_{x \to 0^+} \frac{9}{8}$$
=
$$\lim_{x \to 0^+} \frac{9}{8}$$
=
$$\frac{9}{8}$$
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
$$\lim_{x \to 0^+} \sin{\left(9 x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+}\left(8 x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
=
Let's transform the function under the limit a few
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(9 x \right)}}{\frac{d}{d x} 8 x}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{9 \cos{\left(9 x \right)}}{8}\right)$$
=
$$\lim_{x \to 0^+} \frac{9}{8}$$
=
$$\lim_{x \to 0^+} \frac{9}{8}$$
=
$$\frac{9}{8}$$
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(9*x)\ lim |--------| x->0+\ 8*x /
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
9/8
$$\frac{9}{8}$$
= 1.125
/sin(9*x)\ lim |--------| x->0-\ 8*x /
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right)$$
9/8
$$\frac{9}{8}$$
= 1.125
= 1.125
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{9}{8}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{9}{8}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{\sin{\left(9 \right)}}{8}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{\sin{\left(9 \right)}}{8}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{9}{8}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{\sin{\left(9 \right)}}{8}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = \frac{\sin{\left(9 \right)}}{8}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(9 x \right)}}{8 x}\right) = 0$$
More at x→-oo
The graph