- Source: Vertical tangent
In mathematics, particularly calculus, a vertical tangent is a tangent line that is vertical. Because a vertical line has infinite slope, a function whose graph has a vertical tangent is not differentiable at the point of tangency.
Limit definition
A function ƒ has a vertical tangent at x = a if the difference quotient used to define the derivative has infinite limit:
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
=
+
∞
or
lim
h
→
0
f
(
a
+
h
)
−
f
(
a
)
h
=
−
∞
.
{\displaystyle \lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}={+\infty }\quad {\text{or}}\quad \lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}={-\infty }.}
The graph of ƒ has a vertical tangent at x = a if the derivative of ƒ at a is either positive or negative infinity.
For a continuous function, it is often possible to detect a vertical tangent by taking the limit of the derivative. If
lim
x
→
a
f
′
(
x
)
=
+
∞
,
{\displaystyle \lim _{x\to a}f'(x)={+\infty }{\text{,}}}
then ƒ must have an upward-sloping vertical tangent at x = a. Similarly, if
lim
x
→
a
f
′
(
x
)
=
−
∞
,
{\displaystyle \lim _{x\to a}f'(x)={-\infty }{\text{,}}}
then ƒ must have a downward-sloping vertical tangent at x = a. In these situations, the vertical tangent to ƒ appears as a vertical asymptote on the graph of the derivative.
Vertical cusps
Closely related to vertical tangents are vertical cusps. This occurs when the one-sided derivatives are both infinite, but one is positive and the other is negative. For example, if
lim
h
→
0
−
f
(
a
+
h
)
−
f
(
a
)
h
=
+
∞
and
lim
h
→
0
+
f
(
a
+
h
)
−
f
(
a
)
h
=
−
∞
,
{\displaystyle \lim _{h\to 0^{-}}{\frac {f(a+h)-f(a)}{h}}={+\infty }\quad {\text{and}}\quad \lim _{h\to 0^{+}}{\frac {f(a+h)-f(a)}{h}}={-\infty }{\text{,}}}
then the graph of ƒ will have a vertical cusp that slopes up on the left side and down on the right side.
As with vertical tangents, vertical cusps can sometimes be detected for a continuous function by examining the limit of the derivative. For example, if
lim
x
→
a
−
f
′
(
x
)
=
−
∞
and
lim
x
→
a
+
f
′
(
x
)
=
+
∞
,
{\displaystyle \lim _{x\to a^{-}}f'(x)={-\infty }\quad {\text{and}}\quad \lim _{x\to a^{+}}f'(x)={+\infty }{\text{,}}}
then the graph of ƒ will have a vertical cusp at x = a that slopes down on the left side and up on the right side.
Example
The function
f
(
x
)
=
x
3
{\displaystyle f(x)={\sqrt[{3}]{x}}}
has a vertical tangent at x = 0, since it is continuous and
lim
x
→
0
f
′
(
x
)
=
lim
x
→
0
1
3
x
2
3
=
∞
.
{\displaystyle \lim _{x\to 0}f'(x)\;=\;\lim _{x\to 0}{\frac {1}{3{\sqrt[{3}]{x^{2}}}}}\;=\;\infty .}
Similarly, the function
g
(
x
)
=
x
2
3
{\displaystyle g(x)={\sqrt[{3}]{x^{2}}}}
has a vertical cusp at x = 0, since it is continuous,
lim
x
→
0
−
g
′
(
x
)
=
lim
x
→
0
−
2
3
x
3
=
−
∞
,
{\displaystyle \lim _{x\to 0^{-}}g'(x)\;=\;\lim _{x\to 0^{-}}{\frac {2}{3{\sqrt[{3}]{x}}}}\;=\;{-\infty }{\text{,}}}
and
lim
x
→
0
+
g
′
(
x
)
=
lim
x
→
0
+
2
3
x
3
=
+
∞
.
{\displaystyle \lim _{x\to 0^{+}}g'(x)\;=\;\lim _{x\to 0^{+}}{\frac {2}{3{\sqrt[{3}]{x}}}}\;=\;{+\infty }{\text{.}}}
References
Vertical Tangents and Cusps. Retrieved May 12, 2006.
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- Vertical tangent
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- Tangent
- Vertical and horizontal
- Differentiable function
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- Vertical and horizontal bundles
- Implicit function
- Involute
