Mister Exam
Other calculators:
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How to use it?
- Limit of the function:
-
Limit of ((3+5*x)/(-2+4*x))^(1+3*x)
-
Limit of (4-sqrt(x))/(-16+x)
-
Limit of (-4+3*x)/(8-x+6*x^2)
-
Limit of (-7-2*x^2+3*x^4)/(5+3*x+3*x^4)
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Identical expressions
- (four -sqrt(x))/(- sixteen +x)
- (4 minus square root of (x)) divide by ( minus 16 plus x)
- (four minus square root of (x)) divide by ( minus sixteen plus x)
- (4-√(x))/(-16+x)
- 4-sqrtx/-16+x
- (4-sqrt(x)) divide by (-16+x)
-
Similar expressions
- (4+sqrt(x))/(-16+x)
- (4-sqrt(x))/(-16-x)
- (4-sqrt(x))/(16+x)
Limit of the function (4-sqrt(x))/(-16+x)
The solution
You have entered
[src]
/ ___\
|4 - \/ x |
lim |---------|
x->16+\ -16 + x /
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
Limit((4 - sqrt(x))/(-16 + x), x, 16)
Detail solution
Let's take the limit
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
Multiply numerator and denominator by
$$- \sqrt{x} - 4$$
we get
$$\frac{\frac{4 - \sqrt{x}}{x - 16} \left(- \sqrt{x} - 4\right)}{- \sqrt{x} - 4}$$
=
$$\frac{1}{- \sqrt{x} - 4}$$
=
$$\frac{1}{- \sqrt{x} - 4}$$
The final answer:
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
=
$$\lim_{x \to 16^+} \frac{1}{- \sqrt{x} - 4}$$
=
$$- \frac{1}{8}$$
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
Multiply numerator and denominator by
$$- \sqrt{x} - 4$$
we get
$$\frac{\frac{4 - \sqrt{x}}{x - 16} \left(- \sqrt{x} - 4\right)}{- \sqrt{x} - 4}$$
=
$$\frac{1}{- \sqrt{x} - 4}$$
=
$$\frac{1}{- \sqrt{x} - 4}$$
The final answer:
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
=
$$\lim_{x \to 16^+} \frac{1}{- \sqrt{x} - 4}$$
=
$$- \frac{1}{8}$$
Lopital's rule
We have indeterminateness of type
i.e. limit for the numerator is
$$\lim_{x \to 16^+}\left(4 - \sqrt{x}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 16^+}\left(x - 16\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
=
$$\lim_{x \to 16^+}\left(\frac{\frac{d}{d x} \left(4 - \sqrt{x}\right)}{\frac{d}{d x} \left(x - 16\right)}\right)$$
=
$$\lim_{x \to 16^+}\left(- \frac{1}{2 \sqrt{x}}\right)$$
=
$$\lim_{x \to 16^+} - \frac{1}{8}$$
=
$$\lim_{x \to 16^+} - \frac{1}{8}$$
=
$$- \frac{1}{8}$$
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
$$\lim_{x \to 16^+}\left(4 - \sqrt{x}\right) = 0$$
and limit for the denominator is
$$\lim_{x \to 16^+}\left(x - 16\right) = 0$$
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
=
$$\lim_{x \to 16^+}\left(\frac{\frac{d}{d x} \left(4 - \sqrt{x}\right)}{\frac{d}{d x} \left(x - 16\right)}\right)$$
=
$$\lim_{x \to 16^+}\left(- \frac{1}{2 \sqrt{x}}\right)$$
=
$$\lim_{x \to 16^+} - \frac{1}{8}$$
=
$$\lim_{x \to 16^+} - \frac{1}{8}$$
=
$$- \frac{1}{8}$$
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
$$\lim_{x \to 16^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{8}$$
More at x→16 from the left
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{8}$$
$$\lim_{x \to \infty}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{4}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{4}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = 0$$
More at x→-oo
More at x→16 from the left
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{8}$$
$$\lim_{x \to \infty}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = 0$$
More at x→oo
$$\lim_{x \to 0^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{4}$$
More at x→0 from the left
$$\lim_{x \to 0^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{4}$$
More at x→0 from the right
$$\lim_{x \to 1^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{5}$$
More at x→1 from the left
$$\lim_{x \to 1^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = - \frac{1}{5}$$
More at x→1 from the right
$$\lim_{x \to -\infty}\left(\frac{4 - \sqrt{x}}{x - 16}\right) = 0$$
More at x→-oo
One‐sided limits
[src]
/ ___\
|4 - \/ x |
lim |---------|
x->16+\ -16 + x /
$$\lim_{x \to 16^+}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
-1/8
$$- \frac{1}{8}$$
= -0.125
/ ___\
|4 - \/ x |
lim |---------|
x->16-\ -16 + x /
$$\lim_{x \to 16^-}\left(\frac{4 - \sqrt{x}}{x - 16}\right)$$
-1/8
$$- \frac{1}{8}$$
= -0.125
= -0.125
The graph