Mister Exam
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- Limit of the function:
-
Limit of ((3+2*x)/(7+5*x))^(1+x)
-
Limit of (4+x^2-5*x)/(-16+x^2)
-
Limit of (5-5*x)/(-1+sqrt(x))
- Limit of (-tan(a)+tan(x))/(x-a)
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Identical expressions
- (-tan(a)+tan(x))/(x-a)
- ( minus tangent of (a) plus tangent of (x)) divide by (x minus a)
- -tana+tanx/x-a
- (-tan(a)+tan(x)) divide by (x-a)
-
Similar expressions
- (-tan(a)-tan(x))/(x-a)
- (-tan(a)+tan(x))/(x+a)
- (tan(a)+tan(x))/(x-a)
Limit of the function (-tan(a)+tan(x))/(x-a)
The solution
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/-tan(a) + tan(x)\ lim |----------------| x->a+\ x - a /
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
Limit((-tan(a) + tan(x))/(x - a), x, a)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to a^+}\left(- \tan{\left(a \right)} + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to a^+}\left(- a + x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
=
$$\lim_{x \to a^+}\left(\frac{\frac{\partial}{\partial x} \left(- \tan{\left(a \right)} + \tan{\left(x \right)}\right)}{\frac{\partial}{\partial x} \left(- a + x\right)}\right)$$
=
$$\lim_{x \to a^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to a^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\tan^{2}{\left(a \right)} + 1$$
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
$$\lim_{x \to a^+}\left(- \tan{\left(a \right)} + \tan{\left(x \right)}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to a^+}\left(- a + x\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
=
$$\lim_{x \to a^+}\left(\frac{\frac{\partial}{\partial x} \left(- \tan{\left(a \right)} + \tan{\left(x \right)}\right)}{\frac{\partial}{\partial x} \left(- a + x\right)}\right)$$
=
$$\lim_{x \to a^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\lim_{x \to a^+}\left(\tan^{2}{\left(x \right)} + 1\right)$$
=
$$\tan^{2}{\left(a \right)} + 1$$
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)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to a^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \tan^{2}{\left(a \right)} + 1$$
More at x→a from the left
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \tan^{2}{\left(a \right)} + 1$$
$$\lim_{x \to \infty}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{\tan{\left(a \right)}}{a}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{\tan{\left(a \right)}}{a}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{- \tan{\left(a \right)} + \tan{\left(1 \right)}}{1 - a}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{- \tan{\left(a \right)} + \tan{\left(1 \right)}}{1 - a}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
More at x→-oo
More at x→a from the left
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \tan^{2}{\left(a \right)} + 1$$
$$\lim_{x \to \infty}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{\tan{\left(a \right)}}{a}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{\tan{\left(a \right)}}{a}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{- \tan{\left(a \right)} + \tan{\left(1 \right)}}{1 - a}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right) = \frac{- \tan{\left(a \right)} + \tan{\left(1 \right)}}{1 - a}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
More at x→-oo
One‐sided limits
[src]
/-tan(a) + tan(x)\ lim |----------------| x->a+\ x - a /
$$\lim_{x \to a^+}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
2 1 + tan (a)
$$\tan^{2}{\left(a \right)} + 1$$
/-tan(a) + tan(x)\ lim |----------------| x->a-\ x - a /
$$\lim_{x \to a^-}\left(\frac{- \tan{\left(a \right)} + \tan{\left(x \right)}}{- a + x}\right)$$
2 1 + tan (a)
$$\tan^{2}{\left(a \right)} + 1$$
1 + tan(a)^2