Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of (-64+4^x)/(-3+x)
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Limit of (3-x^2+5*x)/(4*x^7+81*x)
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Limit of (-2+2*x^3+7*x)/(-4-x+3*x^3)
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Limit of (3+2*x)/(2+4*x)
- Graphing y =:
- (3+2*x)/(2+4*x)
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Identical expressions
- (three + two *x)/(two + four *x)
- (3 plus 2 multiply by x) divide by (2 plus 4 multiply by x)
- (three plus two multiply by x) divide by (two plus four multiply by x)
- (3+2x)/(2+4x)
- 3+2x/2+4x
- (3+2*x) divide by (2+4*x)
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Similar expressions
- (3+2*x)/(2-4*x)
- (3-2*x)/(2+4*x)
Limit of the function (3+2*x)/(2+4*x)
The solution
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[src]
/3 + 2*x\ lim |-------| x->1/2+\2 + 4*x/
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
Limit((3 + 2*x)/(2 + 4*x), x, 1/2)
Detail solution
Let's take the limit
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
transform
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{2 \left(2 x + 1\right)}\right) = $$
$$\frac{\frac{2}{2} + 3}{2 \left(1 + \frac{2}{2}\right)} = $$
The final answer:
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right) = 1$$
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
transform
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
=
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{2 \left(2 x + 1\right)}\right) = $$
$$\frac{\frac{2}{2} + 3}{2 \left(1 + \frac{2}{2}\right)} = $$
= 1
The final answer:
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right) = 1$$
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
$$\lim_{x \to \frac{1}{2}^-}\left(\frac{2 x + 3}{4 x + 2}\right) = 1$$
More at x→1/2 from the left
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{1}{2}$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{3}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{3}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{5}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{5}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{1}{2}$$
More at x→-oo
More at x→1/2 from the left
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right) = 1$$
$$\lim_{x \to \infty}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{1}{2}$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{3}{2}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{3}{2}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{5}{6}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{5}{6}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{2 x + 3}{4 x + 2}\right) = \frac{1}{2}$$
More at x→-oo
One‐sided limits
[src]
/3 + 2*x\ lim |-------| x->1/2+\2 + 4*x/
$$\lim_{x \to \frac{1}{2}^+}\left(\frac{2 x + 3}{4 x + 2}\right)$$
1
$$1$$
= 1.0
/3 + 2*x\ lim |-------| x->1/2-\2 + 4*x/
$$\lim_{x \to \frac{1}{2}^-}\left(\frac{2 x + 3}{4 x + 2}\right)$$
1
$$1$$
= 1.0
= 1.0
The graph