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Tangent to a Curve

Tool to calculate the tangent to a curve, to a function, at a given point (infinity close to this point) and to find its tangent line equation as a function of the variable x.

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Tangent to a Curve -

Tag(s) : Geometry

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# Tangent to a Curve

## Tangent Line Equation Calculator

### What is a tangent? (Definition)

In geometry, a tangent to a curve is a straight line that approaches / caresses / touches the curve at this point so as to form an angle equal to 0.

### How to calculate the tangent equation?

The equation of the tangent at $x = a$ is calculated from the equation of the curve $f(x)$, by applying a limit calculation and a derivative calculation.

Calculate the limit $$\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$

If the limit is indeterminate, then there is no tangent at this point (the function is not differentiable in $x = a$)

If the limit is infinite ($+\infty$ or $-\infty$), then the line of equation $x = a$ is a tangent in $a$ (and also an asymptote)

Otherwise (the limit is finite, it has a value) then calculate $f'(x)$ the derivative of $f (x)$ in order to obtain the equation of the tangent in $a$ which is defined by the formula $$y = (x-a) \times f'(a) + f(a)$$

Example: Determinate the tangent equation of $f(x) = x^2$ at the point $x = 1$, first calculate the limit $$\lim_{h \to 0} \frac{(1+h)^2-1^2}{h} = \lim_{h \to 0} \frac{2h+h^2}{h} = \lim_{h \to 0} 2+h = 2$$ this limit is therefore finite, the function is differentiable and its derivative is $f'(x) = 2x$, so the tangent equation is $$y = (x-a) \times f'(a) + f(a) \\ y = (x-1) \cdot 2 \cdot 1 + 1^2 \\ y = 2x-2+1 \\ y = 2x-1$$

If it is already known that the function is differentiable in $a$, then the calculation of the limit is not necessary and the formula can be applied directly.

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