Some questions concerning limits.

How do I sketch a graph when I'm given limit info, for example this

lim f =2(as x approaches 0^-)

lim f =0(as x approaches 4^+)

lim f =0(as x approaches 0^+)

lim f =3 ( as x approaches 4^-)

f(0)=2

f(4)=1

I have some vague idea how to do it, but my sketches do not seem like graphs.

Prove that lim sqrt(x)*e^(sin(pi:x)) =0(as x approaches 0^+)

no idea where to start this, I could graph is on graphing calculator but that is worthless, i cannot use limit laws as e^sin(pi/x) doesnt exist as you cant divide by zero.

Suppose that

x^4<f(x)<x^2

if |X|<1 and

x^2<f(x)<x^4

if |x|>1

find value of limits

lim f(x) as x approaches -1

lim f(x) as x approaches 0

lim f(x) as x approaches 1

no clue about that one

Last one I just wanna make sure I did right,

is there a number c such that

lim ((2x^2+cx-4c)/(x^2+x-6)) (as x approaches 2) exists?

If so find value of c and calculate limit

what i did is a bit of guessing

factor x^2+x-6

x^2+x-6=(x-2)(x+3)

now this is where guessing comes in

2x^2+cx-4c.

D=c^2+32c

so D must be a sqrt of something otherwise i cant see it being factored

let c=4 then D=144 then

2x^2+cx-4c=2x^2+4x-16=(x+4)(x-2)

x-2 is canceled, therefore

(x+4)/(x+3)

rest is obvious.

Forgive me for my wall of text and my poor understanding of math, any help is appreciated!