Mister Exam
Other calculators:
-
How to use it?
- Limit of the function:
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Limit of (1+3*x)^(5/x)
-
Limit of x^2/(-2+sqrt(4+x^2))
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Limit of ((5+x^2-6*x)/(5+x^2-5*x))^(2+3*x)
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Limit of (2+x^2+3*x)/(-4+x^2)
- Graphing y =:
- (1+3*x)^(5/x)
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Identical expressions
- (one + three *x)^(five /x)
- (1 plus 3 multiply by x) to the power of (5 divide by x)
- (one plus three multiply by x) to the power of (five divide by x)
- (1+3*x)(5/x)
- 1+3*x5/x
- (1+3x)^(5/x)
- (1+3x)(5/x)
- 1+3x5/x
- 1+3x^5/x
- (1+3*x)^(5 divide by x)
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Similar expressions
- (1-3*x)^(5/x)
Limit of the function (1+3*x)^(5/x)
The solution
You have entered
[src]
5
-
x
lim (1 + 3*x)
x->0+
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}$$
Limit((1 + 3*x)^(5/x), x, 0)
Detail solution
Let's take the limit
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}$$
transform
do replacement
$$u = \frac{1}{3 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{3}{\frac{1}{x}}\right)^{\frac{5}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15} = e^{15}$$
The final answer:
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}$$
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}$$
transform
do replacement
$$u = \frac{1}{3 x}$$
then
$$\lim_{x \to 0^+} \left(1 + \frac{3}{\frac{1}{x}}\right)^{\frac{5}{x}}$$ =
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}$$
=
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{15 u}$$
=
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15}$$
The limit
$$\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}$$
is second remarkable limit, is equal to e ~ 2.718281828459045
then
$$\left(\left(\lim_{u \to 0^+} \left(1 + \frac{1}{u}\right)^{u}\right)\right)^{15} = e^{15}$$
The final answer:
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}$$
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
[src]
5
-
x
lim (1 + 3*x)
x->0+
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}}$$
15 e
$$e^{15}$$
= 3269017.37247211
5
-
x
lim (1 + 3*x)
x->0-
$$\lim_{x \to 0^-} \left(3 x + 1\right)^{\frac{5}{x}}$$
15 e
$$e^{15}$$
exp(15)
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to 0^-} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}$$
More at x→0 from the left
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}$$
$$\lim_{x \to \infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(3 x + 1\right)^{\frac{5}{x}} = 1024$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(3 x + 1\right)^{\frac{5}{x}} = 1024$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
More at x→-oo
More at x→0 from the left
$$\lim_{x \to 0^+} \left(3 x + 1\right)^{\frac{5}{x}} = e^{15}$$
$$\lim_{x \to \infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
More at x→oo
$$\lim_{x \to 1^-} \left(3 x + 1\right)^{\frac{5}{x}} = 1024$$
More at x→1 from the left
$$\lim_{x \to 1^+} \left(3 x + 1\right)^{\frac{5}{x}} = 1024$$
More at x→1 from the right
$$\lim_{x \to -\infty} \left(3 x + 1\right)^{\frac{5}{x}} = 1$$
More at x→-oo
The graph