Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
- Limit of tan(m*x)/sin(n*x)
-
Limit of sqrt(x+sqrt(x+sqrt(x)))-sqrt(x)
-
Limit of sin(x)/log(1+2*x)
-
Limit of sin(pi*x/3)
-
Identical expressions
- sin(x)/log(one + two *x)
- sinus of (x) divide by logarithm of (1 plus 2 multiply by x)
- sinus of (x) divide by logarithm of (one plus two multiply by x)
- sin(x)/log(1+2x)
- sinx/log1+2x
- sin(x) divide by log(1+2*x)
-
Similar expressions
- (-atan(x)+asin(x))/log(1+2*x^3)
- sin(x)/log(1-2*x)
- sinx/log(1+2*x)
Limit of the function sin(x)/log(1+2*x)
The solution
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[src]
/ sin(x) \ lim |------------| x->0+\log(1 + 2*x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right)$$
Limit(sin(x)/log(1 + 2*x), x, 0)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \log{\left(2 x + 1 \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \log{\left(2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\left(x + \frac{1}{2}\right) \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\frac{1}{2}$$
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)
0/0,
i.e. limit for the numerator is
$$\lim_{x \to 0^+} \sin{\left(x \right)} = 0$$
and limit for the denominator is
$$\lim_{x \to 0^+} \log{\left(2 x + 1 \right)} = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \sin{\left(x \right)}}{\frac{d}{d x} \log{\left(2 x + 1 \right)}}\right)$$
=
$$\lim_{x \to 0^+}\left(\left(x + \frac{1}{2}\right) \cos{\left(x \right)}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\lim_{x \to 0^+}\left(\frac{\cos{\left(x \right)}}{2}\right)$$
=
$$\frac{1}{2}$$
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
One‐sided limits
[src]
/ sin(x) \ lim |------------| x->0+\log(1 + 2*x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/ sin(x) \ lim |------------| x->0-\log(1 + 2*x)/
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{\sin{\left(1 \right)}}{\log{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{\sin{\left(1 \right)}}{\log{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = 0$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = 0$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{\sin{\left(1 \right)}}{\log{\left(3 \right)}}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = \frac{\sin{\left(1 \right)}}{\log{\left(3 \right)}}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{\log{\left(2 x + 1 \right)}}\right) = 0$$
More at x→-oo
The graph