Limits of a Function

Calculus 1 / Ab

738 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
Carlos P.
02:04
Essential Calculus Early Transcendentals

Prove the statement using the $\varepsilon, \delta$ definition of a limit.
$$\lim _{x \rightarrow 9^{-}} \sqrt[4]{9-x}=0$$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
03:44
Essential Calculus Early Transcendentals

Prove the statement using the $\varepsilon, \varepsilon$ definition of a limit.
$$\lim _{x \rightarrow-1.5} \frac{9-4 x^{2}}{3+2 x}=6$$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
01:54
Essential Calculus Early Transcendentals

Prove the statement using the $\varepsilon, \delta$ definition of a limit.
$$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=5$$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
02:48
Essential Calculus Early Transcendentals

Prove the statement using the $\varepsilon, \delta$ definition of a limit.
$\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x-2}=5$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
04:18
Essential Calculus Early Transcendentals

Prove the statement using the $\varepsilon, \delta$ definition of a limit
and illustrate with a diagram like Figure $15 .$
$\lim _{x \rightarrow-2}(3 x+5)=-1$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
04:41
Essential Calculus Early Transcendentals

$ Prove the statement using the $\varepsilon, \delta$ definition of a limit
and illustrate with a diagram like Figure $15 .$
$$\lim _{x \rightarrow 3}\left(1+\frac{1}{3} x\right)=2$$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
04:28
Essential Calculus Early Transcendentals

A crystal growth furnace is used in research to determine
how best to manufacture crystals used in electronic com-
ponents for the space shuttle. For proper growth of the
crystal, the temperature must be controlled accurately by
adjusting the input power. Suppose the relationship is
given by
$T(w)=0.1 w^{2}+2.155 w+20$
where $T$ is the temperature in degrees Celsius and $w$ is the
power input in watts.
(a) How much power is needed to maintain the temperature
at $200^{\circ} \mathrm{C} ?$
(b) If the temperature is allowed to vary from $200^{\circ} \mathrm{C}$ by up
to $\pm 1^{\circ} \mathrm{C},$ what range of wattage is allowed for the input
power?
(c) In terms of the $\varepsilon, \delta$ definition of $\lim _{x \rightarrow a} f(x)=L,$ what
is $x ?$ What is $f(x) ?$ What is $a ?$ What is $L$ ? What value
of $\varepsilon$ is given? What is the corresponding value of $\delta$ ?

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
03:26
Essential Calculus Early Transcendentals

(a) Evaluate $h(x)=(\tan x-x) / x^{3}$ for $x=1,0.5,0.1$
$0.05,0.01,$ and 0.005
(b) Guess the value of $\lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}}$
(c) Evaluate $h(x)$ for successively smaller values of $x$ until
you finally reach 0 values for $h(x)$ . Are you still con-
fident that your guess in part (b) is correct? Explain
why you eventually obtained 0 values. (In Section 3.7
a method for evaluating the limit will be explained.)
(d) Graph the function $h$ in the viewing rectangle $[-1,1]$
by $[0,1] .$ Then zoom in toward the point where the
graph crosses the $y$ -axis to estimate the limit of $h(x)$ as
$x$ approaches $0 .$ Continue to zoom in until you observe
distortions in the graph of $h .$ Compare with the results
of part (c).

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
01:28
Essential Calculus Early Transcendentals

(a) Evaluate the function $f(x)=x^{2}-\left(2^{x} / 1000\right)$ for
$x=1,0.8,0.6,0.4,0.2,0.1,$ and $0.05,$ and guess the
value of
$\lim _{x \rightarrow 0}\left(x^{2}-\frac{2^{x}}{1000}\right)$
(b) Evaluate $f(x)$ for $x=0.04,0.02,0.01,0.005,0.003$
and $0.001 .$ Guess again.

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
01:41
Essential Calculus Early Transcendentals

Use a table of values to estimate the value of the
limit. If you have a graphing device, use it to confirm your
result graphically.
$\lim _{x \rightarrow 0} \frac{9^{x}-5^{x}}{x}$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
01:28
Essential Calculus Early Transcendentals

Use a table of values to estimate the value of the
limit. If you have a graphing device, use it to confirm your
result graphically.
$\lim _{x \rightarrow 1} \frac{x^{6}-1}{x^{10}-1}$

FUNCTIONS AND LIMITS
The Limit of a Function
Bon Z.
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