Mister Exam
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- Limit of the function:
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Limit of (-6+x^2-x)/(6+x^2-5*x)
-
Limit of log(1+x^2)/(-e^(-x)+cos(3*x))
-
Limit of -2+|-2+x|/x
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Limit of (-4-8*x+8*x^2)/(3+8*x)
-
Identical expressions
- five ^x/x
- 5 to the power of x divide by x
- five to the power of x divide by x
- 5x/x
- 5^x divide by x
Limit of the function 5^x/x
The solution
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/ x\
|5 |
lim |--|
x->oo\x /
$$\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right)$$
Limit(5^x/x, x, oo, dir='-')
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to \infty} 5^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 5^{x}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\right)$$
=
$$\infty$$
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)
oo/oo,
i.e. limit for the numerator is
$$\lim_{x \to \infty} 5^{x} = \infty$$
and limit for the denominator is
$$\lim_{x \to \infty} x = \infty$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right)$$
=
$$\lim_{x \to \infty}\left(\frac{\frac{d}{d x} 5^{x}}{\frac{d}{d x} x}\right)$$
=
$$\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\right)$$
=
$$\lim_{x \to \infty}\left(5^{x} \log{\left(5 \right)}\right)$$
=
$$\infty$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{5^{x}}{x}\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{5^{x}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{5^{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{5^{x}}{x}\right) = 5$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{5^{x}}{x}\right) = 5$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{5^{x}}{x}\right) = 0$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{5^{x}}{x}\right) = -\infty$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{5^{x}}{x}\right) = \infty$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{5^{x}}{x}\right) = 5$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{5^{x}}{x}\right) = 5$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{5^{x}}{x}\right) = 0$$
More at x→-oo
The graph