Thread: limits and derivative questions

first problem i'm having is to find what value of the constant c is the function f continuous on (-, )?


i'm not sure how to figure htis one out and all my teacher said was to use the limit laws, kinda puzzled where to start

second problem for me is to find a function g that agrees with f for x a and is continuous at a for:



and




it's clear that both of them have a removable discontinuity but i don't understand when it asks to find a function g for x a

my third problem is as follows:

For all x > 1 the following inequality is true.
Use this inequality to find the limit.

i have no idea as to where to start for this problem, this is the one i'd really like to figure out the most

my next problem is if the tangent line to y = f (x) at (8,6) passes through the point (0,5), find

(a) f (8)
and
(b) f ' (8)

and my last problem is this:
A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and speed when t = 3.
f(t) = t -1 - t

for this problem i thought it was simply plug in 3 for t and you'd get the result for speed however that wasn't the case and as for velocity, i'm guessing i'd have to find the derivative of f(t) but i tried to do it and it's not working out as i thought it would

studying for my test and any advice/tips/help would be appreciated.

thanks

Originally Posted by deltemis

first problem i'm having is to find what value of the constant c is the function f continuous on (-, )?


i'm not sure how to figure htis one out and all my teacher said was to use the limit laws, kinda puzzled where to start

second problem for me is to find a function g that agrees with f for x a and is continuous at a for:



and




it's clear that both of them have a removable discontinuity but i don't understand when it asks to find a function g for x a

my third problem is as follows:

For all x > 1 the following inequality is true.
Use this inequality to find the limit.

i have no idea as to where to start for this problem, this is the one i'd really like to figure out the most

my next problem is if the tangent line to y = f (x) at (8,6) passes through the point (0,5), find

(a) f (8)
and
(b) f ' (8)

and my last problem is this:
A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and speed when t = 3.
f(t) = t -1 - t

for this problem i thought it was simply plug in 3 for t and you'd get the result for speed however that wasn't the case and as for velocity, i'm guessing i'd have to find the derivative of f(t) but i tried to do it and it's not working out as i thought it would

studying for my test and any advice/tips/help would be appreciated.

thanks

For your first question, for the limit to exist, the function must approach the same number from both sides.

In your case, this will happen when

$\displaystyle cx^2 + 2x = x^3 - cx$ at the point $\displaystyle x = 1$.

Thus $\displaystyle c + 2 = 1 - c$

$\displaystyle 2c = -1$

$\displaystyle c = -\frac{1}{2}$.



For the next question, you have the function $\displaystyle f(x) = \frac{x^4 - 1}{x - 1}$

$\displaystyle = \frac{(x^2 - 1)(x^2 + 1)}{x - 1}$

$\displaystyle = \frac{(x - 1)(x + 1)(x^2 + 1)}{x - 1}$.


This function is identical to $\displaystyle g(x) = (x + 1)(x^2+ 1)$ EXCEPT at the point $\displaystyle x = 1$, since $\displaystyle f(x) $ was undefined at $\displaystyle x = 1$.

Can you evaluate $\displaystyle \lim_{x \to 1}f(x)$ now?

Can you try the next one?

For the first problem take the limit of f(x) as x approaches 1 from left and another limit from the right. So you will use both pieces. The you set your results equal to each other and find the value of c.

ahhh thanks guys, didn't realize that first problem was so simple

as for the second problem, it didn't even occur to me to factor

for my second problem part b), it should turn out to be x(x+8) correct? the limit would then be defined but just making sure

Originally Posted by deltemis

first problem i'm having is to find what value of the constant c is the function f continuous on (-, )?


i'm not sure how to figure htis one out and all my teacher said was to use the limit laws, kinda puzzled where to start

second problem for me is to find a function g that agrees with f for x a and is continuous at a for:



and




it's clear that both of them have a removable discontinuity but i don't understand when it asks to find a function g for x a

my third problem is as follows:

For all x > 1 the following inequality is true.
Use this inequality to find the limit.

i have no idea as to where to start for this problem, this is the one i'd really like to figure out the most

my next problem is if the tangent line to y = f (x) at (8,6) passes through the point (0,5), find

(a) f (8)
and
(b) f ' (8)

and my last problem is this:
A particle moves along a straight line with equation of motion s = f(t), where s is measured in meters and t in seconds. Find the velocity and speed when t = 3.
f(t) = t -1 - t

for this problem i thought it was simply plug in 3 for t and you'd get the result for speed however that wasn't the case and as for velocity, i'm guessing i'd have to find the derivative of f(t) but i tried to do it and it's not working out as i thought it would

studying for my test and any advice/tips/help would be appreciated.

thanks

For $\displaystyle \frac{10e^{x}-19}{2e^x} < f(x) < \frac{5\sqrt{x}}{\sqrt{x - 1}}$.

Notice that

$\displaystyle \lim_{x \to \infty}\frac{10e^{x}-19}{2e^x} < \lim_{x \to \infty} f(x) < \lim_{x \to \infty}\frac{5\sqrt{x}}{\sqrt{x - 1}}$.

Evaluate those limits, then it should tell you something about $\displaystyle \lim_{x \to \infty}f(x)$.

ahhh i get it now, squeeze theorem, thanks prove it