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1 answer:
Answer:
cscθ = 5/4, secθ = 5/3
Step-by-step explanation:
First, let's view the drawing that I have attached, with the dot in the bottom left representing the origin. We know that cosecant and secant are the reciprocals of sin and cos respectively, and in order to find sin and cos, we must find the opposite, adjacent, and hypotenuse sides. The hypotenuse is opposite the right angle, equal to √(8²+6²) = 10
Next, we can find sin and cos of θ . sinθ = 8/10=4/5, making cscθ the reciprocal of 4/5, or 5/4
Similarly, cosθ = 6/10=3/5, and secθ = 5/3
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Part A
The notation means that we're approaching x = 2 from the right hand side (aka positive side). This is known as a right hand limit.
So we could start at say x = 2.5 and get closer to 2 by getting to x = 2.4 then to x = 2.3 then 2.2, 2.1, 2.01, 2.001, etc
We don't actually arrive at x = 2 itself. We simply move closer and closer.
Since we're on the positive or right hand side of 2, this means we go with the rule involving x > 2
Therefore f(x) = (x/2) + 1
Plug in x = 2 to find that...
f(x) = (x/2) + 1
f(2) = (2/2) + 1
f(2) = 2
This shows
Then for the left hand limit , we'll involve x < 2 and we go for the first piece. So,
f(x) = 3-x
f(2) = 3-2
f(2) = 1
Therefore,
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Part B
Because this means that the limit
does not exist.
If you are a visual learner, check out the graph below of the piecewise function. Notice the gap or disconnect at x = 2. This can be thought of as two roads that are disconnected. There's no way for a car to go from one road to the other. Because of this disconnect, the limit doesn't exist at x = 2.
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Part C
You'll follow the same type of steps shown in part A.
However, keep in mind that x = 4 is above x = 2, so we'll deal with x > 2 only.
So you'd only involve the second piece f(x) = (x/2) + 1
You should find that f(4) = 3, and that both left and right hand limits equal this value. The left and right hand limits approach the same y value. The limit does exist here. There are no gaps to worry about when x = 4.
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Part D
As mentioned earlier, since , this means the limit
does exist and it's equal to 3.
As x gets closer and closer to 4, the y values are approaching 3. This applies to both directions.
