Mister Exam
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How to use it?
- Limit of the function:
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Limit of (9+3*x^2+4*x)/(7-7*x+3*x^2)
-
Limit of -pi+(1+cos(x))/x
-
Limit of (1-5*x)^(1/x)
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Limit of (1+2/n)^(4*n)
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Identical expressions
- one +x/ two
- 1 plus x divide by 2
- one plus x divide by two
- 1+x divide by 2
-
Similar expressions
- x*(1+x)/(2*sin(x^2))
- ((-1+x)/(2+x))^(2+x)
- 1-x/2
- ((-1+x)/(2+x))^(1+2*x)
- (1+x)*log(1+x)/((2+x)*log(2+x))
Limit of the function 1+x/2
The solution
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/ x\ lim |1 + -| x->oo\ 2/
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right)$$
Limit(1 + x/2, x, oo, dir='-')
Detail solution
Let's take the limit
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{2} + \frac{1}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{2} + \frac{1}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{u + \frac{1}{2}}{u}\right)$$
=
$$\frac{1}{0 \cdot 2} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right) = \infty$$
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right)$$
Let's divide numerator and denominator by x:
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right)$$ =
$$\lim_{x \to \infty}\left(\frac{\frac{1}{2} + \frac{1}{x}}{\frac{1}{x}}\right)$$
Do Replacement
$$u = \frac{1}{x}$$
then
$$\lim_{x \to \infty}\left(\frac{\frac{1}{2} + \frac{1}{x}}{\frac{1}{x}}\right) = \lim_{u \to 0^+}\left(\frac{u + \frac{1}{2}}{u}\right)$$
=
$$\frac{1}{0 \cdot 2} = \infty$$
The final answer:
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right) = \infty$$
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
Other limits x→0, -oo, +oo, 1
$$\lim_{x \to \infty}\left(\frac{x}{2} + 1\right) = \infty$$
$$\lim_{x \to 0^-}\left(\frac{x}{2} + 1\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{2} + 1\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{2} + 1\right) = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{2} + 1\right) = \frac{3}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{2} + 1\right) = -\infty$$
More at x→-oo
$$\lim_{x \to 0^-}\left(\frac{x}{2} + 1\right) = 1$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{x}{2} + 1\right) = 1$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{x}{2} + 1\right) = \frac{3}{2}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{x}{2} + 1\right) = \frac{3}{2}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{x}{2} + 1\right) = -\infty$$
More at x→-oo
The graph