# 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
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
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
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
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
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
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 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 03:26 Essential Calculus Early Transcendentals (a) Evaluate$h(x)=(\tan x-x) / x^{3}$for$x=1,0.5,0.10.05,0.01,$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 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 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 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
1 2 3 4 5 ... 62