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Vertical tangent

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Vertical tangent on the function ƒ(x) at x = c.

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{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{x^{2}}}}}\;=\;\infty .}

Similarly, the function

g ( x ) = x 2 3 {\displaystyle g(x)={\sqrt{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{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{x}}}}\;=\;{+\infty }{\text{.}}}

References

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