# Limits of a Function

### 398 Practice Problems

04:08
Calculus of a Single Variable

In Exercises 63-66, use the definition of limits at infinity to prove the limit.
$$\lim _{x \rightarrow \infty} \frac{2}{\sqrt{x}}=0$$

Applications of Differentiation
Limits at Infinity
25:42
Physics for Scientists and Engineers with Modern Physics

(III) A particle of mass $m$ is projected horizontally at a relativistic speed $v_{0}$ in the $+x$ direction. There is a constant downward force $F$ acting on the particle. Using the definition of relativistic momentum $\vec{\mathbf{p}}=\gamma m \vec{\mathbf{v}}$ and Newton's second law $\vec{\mathbf{F}}=d \vec{\mathbf{p}} / d t,($ a) show that the $x$ and $y$ components of the velocity of the particle at time $t$ are given by
\begin{aligned} v_{x}(t) &=p_{0} c /\left(m^{2} c^{2}+p_{0}^{2}+F^{2} t^{2}\right)^{\frac{1}{2}} \\ v_{y}(t) &=-F c t /\left(m^{2} c^{2}+p_{0}^{2}+F^{2} t^{2}\right)^{\frac{1}{2}} \end{aligned}
where $\quad p_{0}$ is the initial momentum of the particle.
(b) Assume the particle is an electron $\left(m=9.11 \times 10^{-31} \mathrm{kg}\right)$ with $v_{0}=0.50 c \quad$ and $F=1.00 \times 10^{-15} \mathrm{N} .$ Calculate the values of $v_{x}$ and $v_{y}$ of the electron as a function of time $t$ from $t=0$ to $t=5.00 \mu \mathrm{s}$ in intervals of 0.05$\mu \mathrm{s} .$ Graph the values to show how the velocity components change with time during this interval. (c) Is the path parabolic, as it would be in classical mechanics? Explain.

Special Theory of Relativity
Rates of Change and Limits
19:45
Physics for Scientists and Engineers with Modern Physics

A slab of glass with index of refraction $n$ moves to the right with speed $v .$ A flash of light is emitted at point $A$ (Fig. 18) and passes through the glass arriving at point $B$ a distance $\ell$ away. The glass has thickness $d$ in the reference frame where it is at rest, and the speed of light in the glass is $c / n .$ How long does it take the light to go from point $A$ to point $B$ according to an observer at rest with respect to points $A$ and $B ?$ Check your answer for the cases $v=c, v=0$ and $n=1$

Special Theory of Relativity
Rates of Change and Limits
15:32
Physics for Scientists and Engineers with Modern Physics

A pi meson of mass $m_{\pi}$ decays at rest into a muon (mass $m_{\mu} )$ and a neutrino of negligible or zero mass. Show that the kinetic energy of the muon is $K_{\mu}=\left(m_{\pi}-m_{\mu}\right)^{2} c^{2} /\left(2 m_{\pi}\right)$.

Special Theory of Relativity
Rates of Change and Limits
08:19
Physics for Scientists and Engineers with Modern Physics

(II) Calculate the kinetic energy and momentum of a proton $\left(m=1.67 \times 10^{-27} \mathrm{kg}\right)$ traveling $8.15 \times 10^{7} \mathrm{m} / \mathrm{s} .$ By what percentages would your calculations have been in error if you had used classical formulas?

Special Theory of Relativity
Rates of Change and Limits
06:07
Physics for Scientists and Engineers with Modern Physics

(III) In the old West, a marshal riding on a train traveling 35.0 $\mathrm{m} / \mathrm{s}$ sees a duel between two men standing on the Earth 55.0 $\mathrm{m}$ apart parallel to the train. The marshal's instruments indicate that in his reference frame the two men fired simultaneously. (a) Which of the two men, the first one the train passes (A) or the second one (B) should be arrested for firing the first shot? That is, in the gunfighter's frame of reference, who fired first? $(b)$ How much earlier did he fire? (c) Who was struck first?

Special Theory of Relativity
Rates of Change and Limits
05:18
Physics for Scientists and Engineers with Modern Physics

(II) A friend speeds by you in her spacecraft at a speed of 0.760$c .$ It is measured in your frame to be 4.80 $\mathrm{m}$ long and 1.35 $\mathrm{m}$ high. (a) What will be its length and height at rest? (b) How many seconds elapsed on your friend's watch when 20.0 s passed on yours? (c) How fast did you appear to be traveling according to your friend? (d) How many seconds elapsed on your watch when she saw 20.0 s pass on hers?

Special Theory of Relativity
Rates of Change and Limits
02:01
Physics for Scientists and Engineers with Modern Physics

A bowl contains a large number of red, orange, and green jelly beans. You are to make a line of three jelly beans.(a) Construct a table showing the number of microstates that correspond to each macrostate. Then determine the probability of $(b)$ all 3 beans red, and $(c) 2$ greens, 1 orange.

Second Law of Thermodynamics
Rates of Change and Limits
01:26
Physics for Scientists and Engineers with Modern Physics

Thermodynamic processes can be represented not only on $P V$ and $P T$ diagrams; another useful one is a $T S$ (temperature-entropy) diagram. $(a)$ Draw a TS diagram for a Carnot
cycle. $(b)$ What does the area within the curve represent?

Second Law of Thermodynamics
Rates of Change and Limits
04:29
Physics for Scientists and Engineers with Modern Physics

The Stirling cycle, shown in Fig. $27,$ is useful to describe external combustion engines as well as solar-power systems. Find the efficiency of the cycle in terms of the parameters shown, assuming a monatomic gas as the working substance. The processes ab and cd are isothermal whereas bc
and da are at constant volume. How does it compare to the Carnot efficiency?

Second Law of Thermodynamics
Rates of Change and Limits
11:09
Calculus; Graphical, Numerical, Algebraic

Limits and Geometry Let $P \left( a , a ^ { 2 } \right)$ be a point on the parabola $y = x ^ { 2 } , a > 0 .$ Let $O$ be the origin and $( 0 , b )$ the $y$ -intercept of the perpendicular bisector of line segment $O P .$ Find $\lim _ { P \rightarrow O } b$

Limits and Continuity
Rates of Change and Limits
03:41
Calculus; Graphical, Numerical, Algebraic

Controlling Outputs Let $f ( x ) = \sqrt { 3 x - 2 }$
(a) Show that $\lim _ { x \rightarrow 2 } f ( x ) = 2 = f ( 2 )$
(b) Use a graph to estimate values for $a$ and $b$ so that $$1.8 < f ( x ) < 2.2$$ provided $$a < x < b$$
(c) Use a graph to estimate values for $a$ and $b$ so that $$1.99 < f ( x ) < 2.01$$ provided $$a < x < b$$

Limits and Continuity
Rates of Change and Limits
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