Mister Exam
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How to use it?
- Limit of the function:
- Limit of 2016+965*e^(-168*x)*sin(-115+547*x)
- Limit of (1-sqrt(-2+3*x))/(2+x^2-3*x)
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Limit of (1+sin(x))/(1+cos(x))
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Limit of (1+n)^(1/3)/n^(1/3)
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Identical expressions
- (one +sin(x))/(one +cos(x))
- (1 plus sinus of (x)) divide by (1 plus co sinus of e of (x))
- (one plus sinus of (x)) divide by (one plus co sinus of e of (x))
- 1+sinx/1+cosx
- (1+sin(x)) divide by (1+cos(x))
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Similar expressions
- (1-sin(x))/(1+cos(x))
- (1+sin(x))/(1-cos(x))
- (1+sinx)/(1+cosx)
Limit of the function (1+sin(x))/(1+cos(x))
The solution
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/1 + sin(x)\ lim |----------| x->0+\1 + cos(x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right)$$
Limit((1 + sin(x))/(1 + cos(x)), x, 0)
Lopital's rule
There is no sense to apply Lopital's rule to this function since there is no indeterminateness of 0/0 or oo/oo type
The graph
One‐sided limits
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/1 + sin(x)\ lim |----------| x->0+\1 + cos(x)/
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
/1 + sin(x)\ lim |----------| x->0-\1 + cos(x)/
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right)$$
1/2
$$\frac{1}{2}$$
= 0.5
= 0.5
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{1}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = 1$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{\sin{\left(1 \right)} + 1}{\cos{\left(1 \right)} + 1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{\sin{\left(1 \right)} + 1}{\cos{\left(1 \right)} + 1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{1}{2}$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = 1$$
More at x→oo
$$\lim_{x \to 1^-}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{\sin{\left(1 \right)} + 1}{\cos{\left(1 \right)} + 1}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = \frac{\sin{\left(1 \right)} + 1}{\cos{\left(1 \right)} + 1}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} + 1}{\cos{\left(x \right)} + 1}\right) = 1$$
More at x→-oo
The graph