1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Aleks04 [339]
2 years ago
13

How do you find the limit?

Mathematics
1 answer:
coldgirl [10]2 years ago
8 0

Answer:

2/5

Step-by-step explanation:

Hi! Whenever you find a limit, you first directly substitute x = 5 in.

\displaystyle \large{ \lim_{x \to 5} \frac{x^2-6x+5}{x^2-25}}\\

\displaystyle \large{ \lim_{x \to 5} \frac{5^2-6(5)+5}{5^2-25}}\\

\displaystyle \large{ \lim_{x \to 5} \frac{25-30+5}{25-25}}\\

\displaystyle \large{ \lim_{x \to 5} \frac{0}{0}}

Hm, looks like we got 0/0 after directly substitution. 0/0 is one of indeterminate form so we have to use another method to evaluate the limit since direct substitution does not work.

For a polynomial or fractional function, to evaluate a limit with another method if direct substitution does not work, you can do by using factorization method. Simply factor the expression of both denominator and numerator then cancel the same expression.

From x²-6x+5, you can factor as (x-5)(x-1) because -5-1 = -6 which is middle term and (-5)(-1) = 5 which is the last term.

From x²-25, you can factor as (x+5)(x-5) via differences of two squares.

After factoring the expressions, we get a new Limit.

\displaystyle \large{ \lim_{x\to 5}\frac{(x-5)(x-1)}{(x-5)(x+5)}}

We can cancel x-5.

\displaystyle \large{ \lim_{x\to 5}\frac{x-1}{x+5}}

Then directly substitute x = 5 in.

\displaystyle \large{ \lim_{x\to 5}\frac{5-1}{5+5}}\\

\displaystyle \large{ \lim_{x\to 5}\frac{4}{10}}\\

\displaystyle \large{ \lim_{x\to 5}\frac{2}{5}=\frac{2}{5}}

Therefore, the limit value is 2/5.

L’Hopital Method

I wouldn’t recommend using this method since it’s <em>too easy</em> but only if you know the differentiation. You can use this method with a limit that’s evaluated to indeterminate form. Most people use this method when the limit method is too long or hard such as Trigonometric limits or Transcendental function limits.

The method is basically to differentiate both denominator and numerator, do not confuse this with quotient rules.

So from the given function:

\displaystyle \large{ \lim_{x \to 5} \frac{x^2-6x+5}{x^2-25}}

Differentiate numerator and denominator, apply power rules.

<u>Differential</u> (Power Rules)

\displaystyle \large{y = ax^n \longrightarrow y\prime= nax^{n-1}

<u>Differentiation</u> (Property of Addition/Subtraction)

\displaystyle \large{y = f(x)+g(x) \longrightarrow y\prime = f\prime (x) + g\prime (x)}

Hence from the expressions,

\displaystyle \large{ \lim_{x \to 5} \frac{\frac{d}{dx}(x^2-6x+5)}{\frac{d}{dx}(x^2-25)}}\\&#10;&#10;\displaystyle \large{ \lim_{x \to 5} \frac{\frac{d}{dx}(x^2)-\frac{d}{dx}(6x)+\frac{d}{dx}(5)}{\frac{d}{dx}(x^2)-\frac{d}{dx}(25)}}

<u>Differential</u> (Constant)

\displaystyle \large{y = c \longrightarrow y\prime = 0 \ \ \ \ \sf{(c\ \  is \ \ a \ \ constant.)}}

Therefore,

\displaystyle \large{ \lim_{x \to 5} \frac{2x-6}{2x}}\\&#10;&#10;\displaystyle \large{ \lim_{x \to 5} \frac{2(x-3)}{2x}}\\&#10;&#10;\displaystyle \large{ \lim_{x \to 5} \frac{x-3}{x}}

Now we can substitute x = 5 in.

\displaystyle \large{ \lim_{x \to 5} \frac{5-3}{5}}\\&#10;&#10;\displaystyle \large{ \lim_{x \to 5} \frac{2}{5}}=\frac{2}{5}

Thus, the limit value is 2/5 same as the first method.

Notes:

  • If you still get an indeterminate form 0/0 as example after using l’hopital rules, you have to differentiate until you don’t get indeterminate form.
You might be interested in
What is the value of the expression 6x - y when x = 3 and y = 1? (5 points)
Vladimir [108]

Answer:

the answer is 17

Step-by-step explanation:

6×3-1

= 17

6 0
3 years ago
Steven bought five bags of candies for $24.50 total. Which
vladimir2022 [97]

Answer:

4.9

Step-by-step explanation:

24.50 divided by 5 is 4.9

7 0
2 years ago
What is the lateral area of the rectangular prism? Assume the prism is resting on its base. A. 135 in2 B. 270 in2 C. 300 in2 D.
Olegator [25]
You have the length of the sides? If not: a and b is the length of the scratch, h is the height <span>This lateral surface area = 2ah+2bh=2h(a+b)</span>
8 0
3 years ago
When i was 18 years old, my sister was half my age. Now I am 82 years. How old is my sister now?​
iVinArrow [24]

Answer:

41 \:  \:  \: years \:  \:  \: old

Step-by-step explanation:

me \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: sis \\ 18 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: 9 \\ 82 \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  x \\  \\ use \:  \: cross \: \\  multipaction \\ \\  18x = 82 \times 9 \\ 18x = 738 \\  \frac{18x}{18}  =  \frac{738}{18}  \\ x = 41

4 0
3 years ago
Read 2 more answers
A ​$250 suit is marked down by 20​%. Find the sale price.
Olenka [21]
20% off $250, 250 x 0.2= 50
The suit is $50 off, so $250-$50= $200
6 0
3 years ago
Read 2 more answers
Other questions:
  • What is the value of x?
    12·2 answers
  • A rectangular picture measures 6 inches by 9 inches. If a copy machine makes each of the two dimensions 1+1/3 larger, what is th
    7·1 answer
  • Mike bought 5 new baseball trading cards to add to his collection . the next day hos dog ate half of his collection. there are n
    5·1 answer
  • you purchased 10.75 gallons of gasoline at 3.60/gallon. you also bought a cookie (1.25) and windshield wiper fluid at (4.97). th
    5·1 answer
  • Which of the following is not a rate?
    11·1 answer
  • Will give brainliest if you awnser all math questions
    11·1 answer
  • PLZZZZZZZZZZZZZZZZZZZZZZZZZZZ PLZ PLZ HELPPPPPPPP<br> ILL MARK BRAINLEST
    6·2 answers
  • Will get brainily sign if you get it correct with explanation!!!!!
    8·1 answer
  • -x - 4 = 4x - 54 im to lazy to figure it out myself
    6·1 answer
  • A triangular prism has a height of 6 meters and a triangular base with the following dimensions.
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!