Limit Help

[Laucchi]

I'm completely stuck on these problems:

The limit of tan (theta) as theta approaches pi/3


The limit of ((x-2)(x+3))/(x-4) as x approaches 4


The limit of ((x+4)(x-1))/(x-3)

Thank you very much!

 

[mmm4444bot]

What have you tried or thought about, so far?

 

[Laucchi]

Well, for the first one I'm just stumped. I've never studied graphs of tangents.
As for the other two, I've graphed them, but I'm pretty unsure of the answers..

 

[mmm4444bot]

Have you just begun your study of the limit concept?

I ask because the first exercise requires nothing more than knowing how to calculate tan(Pi/3).

How much time has passed, since you studied trigonometry?

 

[mmm4444bot]

The limit of ((x+4)(x-1))/(x-3)

I just noticed that the limit statement above is incomplete. Did they give you a value toward which x is approaching?

 

[mmm4444bot]

I've never studied graphs of tangents.

This will be a big problem for you, in a calculus course.

As for the other two, I've graphed them, but I'm pretty unsure of the answers

f(x) = ((x-2)(x+3))/(x-4)

When you look at your graph of function f's behavior, what is the function doing, both as x approaches the value 4 from below and from above?

 

[Laucchi]

I had a little bit of trigonometry in Pre Calculus, but we only studied the graphs of sines and cosines, and that was a very brief study.

 

[Laucchi]

Yes, I have just started studying limits. I'm homeschooled through online classes, and the class started a few days ago.

 

[Laucchi]

And the last problem is as x approaches 3. My bad.

 

[Laucchi]

f(x) = ((x-2)(x+3))/(x-4)

When you look at your graph of function f's behavior, what is the function doing, both as x approaches the value 4 from below and from above?

The lines are getting closer and closer to 4, but never touching it.

 

[mmm4444bot]

The lines are getting closer and closer to 4, but never touching it.

Best not to describe this function's behavior as "lines". The graph of function f is a "curve".

As x approaches 4 from below, the function value is approaching negative infinity.

As x approaches 4 from above, the function value is approaching positive infinity.

In order for a limit to exist, the function must be approaching a fixed value, and that value must be the same from either direction.

The function in your second exercise does not approach the same, fixed value, as x approaches 4 from either direction, so that limit does not exist.

What do you think about the third exercise, now?

 

[mmm4444bot]

I had a little bit of trigonometry in Pre Calculus, but we only studied the graphs of sines and cosines, and that was a very brief study.

Sounds like those were on-line courses, too.

On-line courses -- in general -- skip a bunch of stuff.

You will have to make up for these shortcomings.

 

[Laucchi]

What do you think about the third exercise, now?

As x approaches 3 from below, the function value is approaching negative infinity.
As x approaches 3 from above, the function value is approaching positive infinity.
So..the limit does not exist.

 

[mmm4444bot]

That's correct. The only limit that exists is the first one.

If you google y=tan(x) you will see an interactive graph at the top of the search results.

Pi/3 is about 1.05

Hover your mouse pointer back-and-forth to confirm that tan(x) approaches the same, fixed value, as x approaches Pi/3 from either direction.

The tangent of Pi/3 is the square root of 3, so that's the limit.

I'm not sure how you're supposed to figure that out, having taken a trig course that skips the tangent function.

tan(x) = sin(x)/cos(x)

If you know sin(Pi/3) and cos(Pi/3), then you may do the algebra to simplify the ratio tan(Pi/3).

 

[Laucchi]

I've never used Google Graphs before, but that was really useful.
Thank you so much for your help! Thank you for making me actually think about the problems instead of just giving me the answers. This will definitely help with the quiz I have to take tomorrow.
Thank you!!!

 

[mmm4444bot]

One is pleased to be of service.

 

[JeffM]

There are some very useful limit laws. Here are some of them:

\(\displaystyle \displaystyle \lim_{x \rightarrow a}g(x) = b\ and\ \lim_{x \rightarrow a}h(x) = c \implies \lim_{x \rightarrow a}\left\{g(x) + h(x)\right\} = b + c.\)

\(\displaystyle \displaystyle \lim_{x \rightarrow a}g(x) = b\ and\ \lim_{x \rightarrow a}h(x) = c \implies \lim_{x \rightarrow a}\left\{g(x) * h(x)\right\} = b * c.\)

\(\displaystyle \displaystyle \lim_{x \rightarrow a}g(x) = b\ and\ \lim_{x \rightarrow a}h(x) = c \ne 0 \implies \lim_{x \rightarrow a}\left\{\dfrac{g(x)}{h(x)}\right\} = \dfrac{b}{c}.\)

\(\displaystyle \displaystyle \lim_{x \rightarrow a}g(x) = b \ne 0\ and\ \lim_{x \rightarrow a}h(x) = c = 0 \implies \lim_{x \rightarrow a}\left\{\dfrac{g(x)}{h(x)}\right\}\ does\ not\ exist.\)

\(\displaystyle \displaystyle \lim_{x \rightarrow a}g(x) = b = 0\ and\ \lim_{x \rightarrow a}h(x) = c = 0 \implies \lim_{x \rightarrow a}\left\{\dfrac{g(x)}{h(x)}\right\}\ may\ or\ may\ not\ exist.\) This situation requires special rules.

These rules would have let you determine quickly whether or not limits existed. In your tangent case, they would also have told you how to calculate the limit of the tangent from the limits of the sine and cosine.