Example 11. Find $\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}$.
Solution. Apply L'Hôpital's rule:

$$
\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}=\lim _{x \rightarrow \infty} \frac{1-\cos x}{1+\cos x}
$$

In the right-hand side of the last equality, the limit does not exist, so L'Hôpital's rule is not applicable here.

The specified limit can be found directly:

$$
\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}=\lim _{x \rightarrow \infty} \frac{1-\frac{\sin x}{x}}{1+\frac{\sin x}{x}}=1
$$

## Problems

Find the limits:

1. $\lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x-\sin x}$.
2. $\lim _{x \rightarrow 0} \frac{\frac{\pi}{x}}{\cot \frac{\pi x}{2}}$.
3. $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 5 x}$.
4. $\lim _{x \rightarrow 0}(1-\cos x) \cot x$.
5. $\lim _{x \rightarrow 1}(1-x) \tan \frac{\pi x}{2}$.
6. $\lim _{x \rightarrow 1}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)$.
6. $\lim _{x \rightarrow 3}\left(\frac{1}{x-3}-\frac{5}{x^{2}-x-6}\right)$.
7. $\lim _{x \rightarrow 0} x^{x}$.
8. $\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)^{\tan x}$.
9. $\lim _{x \rightarrow 0}(\cot x)^{\sin x}$.
10. $\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}\right)^{\frac{1}{1-\cos x}}$.
11. $\lim _{x \rightarrow+\infty}\left(\frac{\pi}{2}-\arctan x\right)^{\frac{1}{\ln x}}$.

## Answers

1.3. 2. $\frac{\pi^{2}}{2}$. 3.5. 4.0.5. $\frac{2}{\pi}$. 6. $\frac{1}{2}$. 7. $\frac{1}{5}$. 8.1. Hint. Let $x^{x}=y$, take the logarithm of this equation, and take the limit. 9.1. 10. 1. 11. $\frac{1}{\sqrt[3]{e}}$. 12. $\frac{1}{e}$.

## §3.2. Tangent and Normal to a Plane Curve. Angle Between Curves. Curvature of a Plane Curve. Velocity and Acceleration

The tangent to a curve $y=f(x)$ at its point $M_{0}\left(x_{0}, y_{0}\right)$ is the limiting position $M_{0} T$ of the secant $M M_{0}$, as the point $M$ approaches $M_{0}$ along the given curve (Fig. 3.1).

The equation of the tangent to the curve $y=f(x)$ at the point $M_{0}\left(x_{0}, y_{0}\right)$ is:

$$
y-y_{0}=f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)
$$

The normal to the curve at the point $M_{0}\left(x_{0}, y_{0}\right)$ is the line passing through $M_{0}$ and perpendicular to the tangent at that point.

The equation of the normal to the curve $y=f(x)$ at the point $M_{0}\left(x_{0}, y_{0}\right)$ is:

$$
y-y_{0}=-\frac{1}{f^{\prime}\left(x_{0}\right)}\left(x-x_{0}\right)
$$

The curvature of a curve at its point $M$ is the limit of the absolute value of the ratio of the angle $\Delta \alpha$ between the tangents at points $M$ and $N$ of the curve to the length of the arc $\breve{M N}=\Delta s$, as $N \rightarrow M$ (Fig. 3.2), i.e.,

$$
k=\lim _{\Delta s \rightarrow 0}\left|\frac{\Delta \alpha}{\Delta s}\right|
$$

where the angle $\alpha$ is expressed in radians.

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-108.jpg?height=412&width=502&top_left_y=240&top_left_x=57)

Fig. 3.1

![](https://cdn.mathpix.com/cropped/2024_05_22_7ef2114a321cf223ba99g-108.jpg?height=412&width=492&top_left_y=244&top_left_x=596)

Fig. 3.2

The curvature of the curve $y=f(x)$ is calculated by the formula

$$
k=\frac{\left|y^{\prime \prime}\right|}{\left(1+y^{\prime 2}\right)^{\frac{3}{2}}}
$$

If the curve is given parametrically by $x=\varphi(t)$, $y=\varphi(t)$, then its curvature is determined by the formula

$$
k=\frac{\left|x^{\prime} y^{\prime \prime}-x^{\prime \prime} y^{\prime}\right|}{\left(x^{\prime 2}+y^{\prime 2}\right)^{\frac{3}{2}}}
$$

The radius of curvature of a curve at a point is the reciprocal of its curvature at that point. If the radius of curvature is denoted by $R$, then

$$
R=\frac{1}{k}
$$

The circle of curvature (osculating circle) of a curve at its point $M$ is the limiting position of the circle passing through the point $M$ and two other points $L$ and $N$ of the curve, as $L \rightarrow M$ and $N \rightarrow M$.

The radius of the circle of curvature is equal to the radius of curvature of the curve at the corresponding point.

The center of the circle of curvature is called the center of curvature of the curve at the given point $M$. The coordinates $X$ and $Y$ of the center of curvature of the curve $y=f(x)$ are calculated by the formulas:

$$
X=x-\frac{y^{\prime}\left(1+y^{\prime 2}\right)}{y^{\prime \prime}} ; Y=y+\frac{1+y^{\prime 2}}{y^{\prime \prime}}
$$

The evolute of a curve is the geometric locus of its centers of curvature.

Formulas (3.6) give the parametric equations of the evolute with parameter $x$ or $y$.

If the curve is given parametrically, then the equations of its evolute are:

$$
X=x-y^{\prime} \frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime} y^{\prime \prime}-y^{\prime} x^{\prime \prime}} ; Y=y+x^{\prime} \frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime} y^{\prime \prime}-y^{\prime} x^{\prime \prime}}
$$

The involute of a curve is a curve for which the given curve is the evolute.