Find the missing side lengths. Leave your answer as radicals in simplest form.

Alex Ar [27]

When we approach limits, we are finding values that are infinitesimally approaching this x-value. Essentially, we consider the approximate location that this root or limit appears. This is essential when it comes to taking Calculus, and finding the limit or rate of change of a function.

When we are attempting limits questions, there are several tests we attempt first.

1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
$\lim_{x \to 0} (\frac{sinx}{x}) = 1$
$\lim_{x \to 0} (\frac{tanx}{x}) = 1$
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.

For example:

1) $\lim_{x \to 0}\frac{\sqrt{x} - 5}{x - 25}$

We can do this using the first and second method.
<em>Method 1: Direct evaluation:</em>

Substitute x = 0 to the function.
$\frac{\sqrt{0} - 5}{0 - 25}$
$= \frac{-5}{-25}$
$= \frac{1}{5}$

<em>Method 2: Rearranging the function
</em>
We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.

$\lim_{x \to 0}\frac{(\sqrt{x} - 5)}{(\sqrt{x} - 5)(\sqrt{x} + 5)}$
$= \lim_{x \to 0}\frac{1}{(\sqrt{x} + 5)}}$
$= \frac{1}{5}$

Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.

8 0

3 years ago

Find the measure indicated. Assume that the lines which appear to be tangent are tangent. PART 3

ludmilkaskok [199]

I'll do problems 13 and 14 to get you started.

=====================================================

Problem 13

<h3>Answer: 32 degrees</h3>

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Explanation:

Refer to the diagram below for problem 13. Since BD is a tangent of the circle, this means angle ABD is 90 degrees, and furthermore x+y = 90. This solves to y = 90-x.

If we focus on triangle ABC, then we can add the interior angles and set the sum equal to 180

A+B+C = 180

z+y+y = 180

z+2y = 180

z+2(90-x) = 180 .... plug in y = 90-x

z+180-2x = 180

z-2x = 180-180

z-2x = 0

z = 2x

We'll use this later.

Now move onto triangle ABD. This is a right triangle due to angle ABD being 90 degrees. The acute angles z and 26 are complementary, meaning they add to 90. So,

z+26 = 90

z = 90-26

z = 64

Now plug this into z = 2x and solve for x

z = 2x

64 = 2x

2x = 64

x = 64/2

x = 32

=====================================================

Problem 14

<h3>Answer: 31 degrees</h3>

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Explanation:

Refer to the image shown below for problem 14. Like before, I've added various labels to it to help find the angle we're after (the red angle x).

Note how I've drawn a dashed line from point A to point D. This splits up quadrilateral ABCD into two triangles ABD and ACD. These triangles can be shown to be congruent using the HL (hypotenuse leg) theorem.

The congruent triangles also implies that angle DBC and angle BCD are the same measure (both are shown in green as the variable y).

Focus on triangle BCD. It has interior angles of: y, y and 62. Add them up, set the sum equal to 180.

y+y+62 = 180

2y+62 = 180

2y = 180-62

2y = 118

y = 118/2

y = 59

As the image attachment mentions, the angles x and y add to 90 degrees. This is because angle ABD is 90 degrees, due to segment BD being tangent to the circle. So x+y = 90 leads to x = 90-y, and therefore,

x = 90-y

x = 90-59

x = 31

Side note: it is not a coincidence that we ended up with half of the value of 62.

3 0

2 years ago

Tell whether each statement is always (A), sometimes (S), or never (N) true.

AnnZ [28]

N an acute angle is 90 degrees or less and 90 + 90 is 180 and an obtuse angle is always more than 90 and always less than 180

7 0

3 years ago

Damon borrowed money from his grandmother 3 years ago and agreed to pay her 4% simple annual interest. At the end of 3 years, he

sergeinik [125]

Answer:

200

Step-by-step explanation:

200×4%=8

8×3=24

The original amount is $200

6 0

3 years ago

Read 2 more answers

What is 15% of 60<br> AMOUNT BASE AND PERCENTAGE

Neko [114]

Answer:

The 15% of 60 is 9 with amount base and percentage

3 0

2 years ago