# How do you do the following questions?

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1.1 What is the value of #lim_(x-> oo) (5x)/(x + 3)# ?

1.2 What is the value of #lim_(x-> 0) (e^x -1 - x)/x^2# ?

3.1 Suppose that the volume of water in a tank is given by #W = t^3/3 - t^2# , where #t# is in minutes. The tap needs to be turned off when the volume is increasing at #15# cubic meters per minute. After how many minutes should the tap be turned off?

5.2 What is the value of #int x/(x +1) dx# ?

5.3 What is the value of #int secx(secx+ tanx)dx# ?

1.1 What is the value of

1.2 What is the value of

3.1 Suppose that the volume of water in a tank is given by

5.2 What is the value of

5.3 What is the value of

##### 2 Answers

**3** .

Call the function

We are asked to find at what time the water is entering the tank at 15 litres per minute. Since

#15 = 3t^2 - 2t#

#0 = 3t^2 - 2t - 15#

#t = (-(-2) +- sqrt((-2)^2 - 4 * 3 * -15))/(2 *3)#

#t = (2 +- sqrt(184))/6#

#t = (2 +- 2sqrt(46))/6#

#t =(1 +- sqrt(46))/3#

There will be one negative solution and one positive solution. The positive solution is the only acceptable one. An approximation for

Thus, the water should be turned off after

**5.2**

We use partial fractions to compute this integral.

#int x/(x + 1)dx#

This will have a partial fraction decomposition of the form

#A/1 + B/(x + 1) = x/(x + 1)#

#A(x + 1) + B = x#

#Ax + A + B = x#

#(A)x + (A + B) =x#

We now have a system of equations

Solve to get

#int 1 -1/(x + 1)dx#

This is separable.

#int 1dx -int1/(x + 1)dx#

#x - ln|x + 1| + C#

**5.3**

We rewrite in terms of sine and cosine.

#int 1/cosx(1/cosx + sinx/cosx)dx#

#int 1/cosx((1 + sinx)/cosx)dx#

#int(1 + sinx)/cosxdx#

#int 1/cosx + sinx/cosx dx#

#int 1/cosx dx + int sinx/cosx dx#

For the second integral, we make the substitution

#int 1/cosx dx + int sinx/u * (du)/(-sinx)#

#int 1/cosxdx + int -1/u du#

The integral

#ln|secx + tanx| - ln|cosx| + C#

Hopefully this helps!

1.1 & 1.2 below

#### Explanation:

**1.1**

Divide numerator and denominator by x:

**1.2**

This is

Instead we use the definition/ Taylor Expansion: