Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of -2+x^3+6*x
-
Limit of ((-2+x)/x)^(2*x)
-
Limit of -(1-cos(2*x))*sin(x)/x+tan(x)
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Limit of (-1+tan(x)^(1/3))/(-1+log(x)/log(pi/4))
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Identical expressions
- (- two + two ^x)/(- one +x)
- ( minus 2 plus 2 to the power of x) divide by ( minus 1 plus x)
- ( minus two plus two to the power of x) divide by ( minus one plus x)
- (-2+2x)/(-1+x)
- -2+2x/-1+x
- -2+2^x/-1+x
- (-2+2^x) divide by (-1+x)
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Similar expressions
- (-2-2^x)/(-1+x)
- (2+2^x)/(-1+x)
- (-2+2^x)/(1+x)
- (-2+2^x)/(-1-x)
Limit of the function (-2+2^x)/(-1+x)
The solution
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[src]
/ x\
|-2 + 2 |
lim |-------|
x->1+\ -1 + x/
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right)$$
Limit((-2 + 2^x)/(-1 + x), x, 1)
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 1^+}\left(2^{x} - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(2^{x} - 2\right)}{\frac{d}{d x} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2^{x} \log{\left(2 \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2 \log{\left(2 \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2 \log{\left(2 \right)}\right)$$
=
$$2 \log{\left(2 \right)}$$
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
$$\lim_{x \to 1^+}\left(2^{x} - 2\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 1^+}\left(x - 1\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right)$$
=
$$\lim_{x \to 1^+}\left(\frac{\frac{d}{d x} \left(2^{x} - 2\right)}{\frac{d}{d x} \left(x - 1\right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2^{x} \log{\left(2 \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2 \log{\left(2 \right)}\right)$$
=
$$\lim_{x \to 1^+}\left(2 \log{\left(2 \right)}\right)$$
=
$$2 \log{\left(2 \right)}$$
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]
/ x\
|-2 + 2 |
lim |-------|
x->1+\ -1 + x/
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right)$$
2*log(2)
$$2 \log{\left(2 \right)}$$
= 1.38629436111989
/ x\
|-2 + 2 |
lim |-------|
x->1-\ -1 + x/
$$\lim_{x \to 1^-}\left(\frac{2^{x} - 2}{x - 1}\right)$$
2*log(2)
$$2 \log{\left(2 \right)}$$
= 1.38629436111989
= 1.38629436111989
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 1^-}\left(\frac{2^{x} - 2}{x - 1}\right) = 2 \log{\left(2 \right)}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right) = 2 \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{2^{x} - 2}{x - 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2^{x} - 2}{x - 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2^{x} - 2}{x - 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{2^{x} - 2}{x - 1}\right) = 0$$
More at x→-oo
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2^{x} - 2}{x - 1}\right) = 2 \log{\left(2 \right)}$$
$$\lim_{x \to \infty}\left(\frac{2^{x} - 2}{x - 1}\right) = \infty$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2^{x} - 2}{x - 1}\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2^{x} - 2}{x - 1}\right) = 1$$
More at x→0 from the right
$$\lim_{x \to -\infty}\left(\frac{2^{x} - 2}{x - 1}\right) = 0$$
More at x→-oo
The graph