12.1 Finding limits: numerical and graphical approaches (Page 4/10)

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Using the graph of the function $y = f (x)$ shown in [link] , estimate the following limits.

Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)

a. 0; b. 2; c. does not exist; d. $- 2;$ e. 0; f. does not exist; g. 4; h. 4; i. 4

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Finding a limit using a table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of $x$ approach $a$ from both sides. Then we determine if the output values get closer and closer to some real value, the limit $L .$

Let’s consider an example using the following function:

\lim_{x \to 5} (\frac{x^{3} - 125}{x - 5})

To create the table, we evaluate the function at values close to $x = 5.$ We use some input values less than 5 and some values greater than 5 as in [link] . The table values show that when $x > 5$ but nearing 5, the corresponding output gets close to 75. When $x > 5$ but nearing 5, the corresponding output also gets close to 75.

Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.

Because

\lim_{x \to 5^{-}} f (x) = 75 = \lim_{x \to 5^{+}} f (x),

then

\lim_{x \to 5} f (x) = 75.

Remember that $f (5)$ does not exist.

How To

Given a function $f,$ use a table to find the limit as $x$ approaches $a$ and the value of $f (a),$ if it exists.

Choose several input values that approach $a$ from both the left and right. Record them in a table.
Evaluate the function at each input value. Record them in the table.
Determine if the table values indicate a left-hand limit and a right-hand limit.
If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
Replace $x$ with $a$ to find the value of $f (a) .$

Finding a limit using a table

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

\lim_{x \to 0} (\frac{5 \sin (x)}{3 x})

We can estimate the value of a limit, if it exists, by evaluating the function at values near $x = 0.$ We cannot find a function value for $x = 0$ directly because the result would have a denominator equal to 0, and thus would be undefined.

f (x) = \frac{5 \sin (x)}{3 x}

We create [link] by choosing several input values close to $x = 0,$ with half of them less than $x = 0$ and half of them greater than $x = 0.$ Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

The table values indicate that when $x < 0$ but approaching 0, the corresponding output nears $\frac{5}{3} .$

When $x > 0$ but approaching 0, the corresponding output also nears $\frac{5}{3} .$

Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.

Because

\lim_{x \to 0^{-}} f (x) = \frac{5}{3} = \lim_{x \to 0^{+}} f (x),

then

\lim_{x \to 0} f (x) = \frac{5}{3} .

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Q&A

Is it possible to check our answer using a graphing utility?

Yes. We previously used a table to find a limit of 75 for the function $f (x) = \frac{x^{3} - 125}{x - 5}$ as $x$ approaches 5. To check, we graph the function on a viewing window as shown in [link] . A graphical check shows both branches of the graph of the function get close to the output 75 as $x$ nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching $x = 5$ we may numerically observe the corresponding outputs getting close to $75.$

Graph of an increasing function with a discontinuity at (5, 75)

Try It

Numerically estimate the limit of the following function by making a table:

\lim_{x \to 0} (\frac{20 \sin (x)}{4 x})

$\lim_{x \to 0} (\frac{20 \sin (x)}{4 x}) = 5$

Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.

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Q&A

Is one method for determining a limit better than the other?

No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

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BenJay

Method

I am eliacin, I need your help in maths

Rood

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Sir

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6

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