limits!!??

Question

the limit as y->36 of (36-y)/(6-(sqrt(y))



and the limit as x->3 of 7arctan(x-2)


ive tried the first one about 5 times and no idea how to do the 2nd... thanks!

Answer 1

lim [36 - y] / [6 - sqrt(y)]
y -> 36

Your first step is to multiply top and bottom by the conjugate of the bottom. The conjugate of (a - b) is (a + b), and when conjugates are multiplied together, they give you a difference of squares,
a^2 - b^2.

In our case, the conjugate is [6 + sqrt(y)], so multiplying top and bottom by that, we get

lim { [36 - y][6 + sqrt(y) } / { [6 - sqrt(y)][6 + sqrt(y)] }
y -> 36

Make the denominator into a difference of squares. Note that
sqrt(y) squared is just y.

lim { [36 - y][6 + sqrt(y) } / [36 - y]
y -> 36

Now, cancel the (36 - y) on the top and bottom.

lim [6 + sqrt(y)]
y -> 36

And at this point, we can safely plug in y = 36 to solve the limit.

6 + sqrt(36) = 6 + 6 = 12

2)

lim 7arctan(x - 2)
x -> 3

We can safely plug in x = 3.

7arctan(3 - 2) = 7arctan(1)

Where is tan equal to 1? Remember arctan(1) is the same as as solving tan(x) = 1, but with the restriction that -pi/2 < x < pi/2. tan is equal to 1 at the point pi/4, so our answer is

7(pi/4) = 7pi/4

Answer 2

mutiple the numberator and denominator by the conjugate, which is the the same as the denominator but the sign is switched

(36-y)(6+sqrt(y)) / ((6-sqrt(y))(6 + sqrt(y))

(36-y)(6+sqrt(y)) / (( 36 - y))

Cancel out (36 - y) from top and bottom and you're left with:

6 + sqrt(y)

now you can plug in 6
your final answer is:

[[12]]

I have no idea how to do number two. I've never seen that "arc" thing before.

Answer 3

36-y =((6-sqrt y)*(6+sqrt y) So your expression can be simplified and is 6+sqrt y => to 12
If x=> 3 Arctan(x-2 ) => to Arctan(1). So Which angle has a tangent= 1 .The answer pi/4 .

So your limit is 7*pi/4

Answer 4

limit as y-> 36 of (36-y) / (6-sqrt(y))

Factor the numerator (top):

= limit as y-> 36 of (6-sqrt(y)) * (6+sqrt(y)) / (6-sqrt(y))

= limit as y -> 36 of (6+sqrt(y))

= limit as y -> 36 of (6 + 6)

= 12