Intervals and plotting functions

Question

I am having a problem with this task. I am asked to plot two functions:

$f_1(x)=\dfrac{x^2-7}{x+\sqrt{7}}$ and $f_2(x)=\dfrac{x^2-7}{x+2.645751311}$

and i am asked to plot these functions with x in the interval [-2.6458 .. -2.6457] and describe what i am seeing. I get this output:

I don't have enough reputation to post images but i get these outputs for function 1 and 2:

https://i.stack.imgur.com/14xN6.png

https://i.stack.imgur.com/PjQ04.png

What am i supposed to be seeing? I see that $f_1$ seems pretty normal but $f_2$ seems to do a wierd cross.

Answer 1

The plots are correct. Note that the function $f_1$ can be transformed: $$f_1(x)=\dfrac{x^2-7}{x+\sqrt{7}} = \dfrac{(x-\sqrt7)(x+\sqrt7)}{x+\sqrt{7}} = x-\sqrt7$$

Hence here the limit for $x \to -\sqrt7$ is well defined for $f_1$. For $f_2$ this does not work, even though $2.645751311$ is very close to $\sqrt7$. This explains the "broken" graph for $f_2$.

Answer 2

The two functions give you different graphs, although their formulas are pretty close. Why? This is what the problem asks you to figure out.

Try to factor function $f_1$ and see if cancellation occurs to simplify it. Notice that a function usually goes to $\pm \infty$ at the point where the denominator equals zero. This happens for function $f_2$. What is special about function $f_1$?