Limits of Function

SBS1984E SG

Hi guys, just started UNI and I need some help to keep my brain going.

Anyone knows how to compute the following limit for each function? The lecture notes aren’t really helping me.

Compute LIM h_> 0 [f(2+h) -f(2) / h]

a. f(x) = sqRT(x) -2

b. f(x) = MOD -3

c. f(x) = 2 / (x-1)

dreamer87 SG

f(x) = sqrt(x) - 2

lim h -> 0, (sqrt(2+h) -2 - (sqrt(2) - 2))/h

By L hospital Rule, lim h->0 1/(2sqrt(2+h) = 1/2sqrt(2).

The rest i think quite similar. It look like year 1 stuff.

wee_ws SG

Hi,

L Hopital's rule is probably an advanced concept at this stage, we could go around it by the method below.

lim h -> 0, [ (sqrt(2+h) - 2 - (sqrt(2) - 2) ] / h

= lim h -> 0, [ sqrt(2+h) - sqrt(2) ] / h

Now we multiply top and bottom by [ sqrt(2 + h) + sqrt(2) ] and this simplifies to

lim h -> 0, 1 / 2sqrt(2)

= 1 / 2sqrt(2).

In doing this limits exercise, we are finding out from first principles the derivative of f at the point where x = 2.

Thanks!

Cheers,
Wen Shih

Kang Kar SG

Hello, good to have this topic. Have question to consult also.

Qn: Find the deriative of the following functions by the limit of theorem or the definition of deriative in terms of x. Check whether the function is differentiable at given point.

a) Price demand funtion p(x) = 6/x at x=0 and x=1

b) Revenue R(x) = 2x^3+ 1 at x=0 and x=1.5

Qn: Revenue (in $) from the sale of x ferry seats is given by:
R(x) = 60x - 0.025x^2 (0 <= x <= 2400)

Find average change in revenue if production changed from 1000 ferry seats to 1050 ferry seats.

Do I need to differentiate the formula given before doing?

Quite rusty in them. Thank you

wee_ws SG

Hi,

Q1 To check whether a function is differentiable at a given point, we determine if

each of p'(0), p'(1), R'(0), R'(1.5) exists.

Q2 Average change = change in R / change in x.

If differentiation is to be used, we will then be interested in the instantaneous rate of change, e.g., R'(x) refers to the instantaneous rate of change at the point x.

Thanks!

Cheers,
Wen Shih

Kang Kar SG

Hello adding on, have another question to ask.

It is from the question asked previously:

Qn: Find the deriative of the following functions by the limit of theorem or the definition of deriative in terms of x. Check whether the function is differentiable at given point.

a) Price demand funtion p(x) = 6/x at x=0 and x=1

b) Revenue R(x) = 2x^3+ 1 at x=0 and x=1.5

For part B), I am stuck with doing the limit of theorem (Formula: {[f(x+h) - f(x)] / h}

[(2x+h)^3+1] - (2x^3 + 1)
h

Do I require to spilt the x^3 into x^2?

Thank you

wee_ws SG

Hi,

The correct expression for the numerator should be

[ 2(x + h)^3 + 1 ] - (2x^3 + 1).

Then you expand and simplify to the result

lim, h -> 0 [ 6x^2 + 6hx + 2h^2 ].

Thanks!

Cheers,
Wen Shih

Kang Kar SG

Hello have more questions to ask.

This time round is regarding deriatives of exponential functions

Find the deriative from the following:

- y' or dy/dx if y = log2x/(1+x2)
Have used the quotient rule but did not manage to get the answer.

- dy/dt if y = tlnt/e^t
Do I use product or quotient rule?

Use the implicit differentiation to find y' and evaluate y' at indicated point:
xlny+2y=2x^3 at (1,1)
Did I do correctly for the differentiation using product rule?
x(1/x)+lny+2-6x^2

Regarding Elasticity in Demand
price - demand equation of product is: 0.02x + p =60

- For which values of p is demand elastic? and Inelastic?
- For which values of p is revenue increasing? and decreasing?

From what I know of:
x = 3000-50p
f(p) = 3000-50p
f'(p) = -50
E(p) = p/(60-p)

Please advise, thank you :)