Are these two graphs touching or intersecting?

Question

Do the curves $x^2 - 1$ and $2^x$ touch or intersect at $x=3$? Both have same values ($=8$) but values of their derivatives at $x=3$ are different. If two curves touch shouldn't their tangents have same slopes? Online graph calculators show the two curves as touching.

Answer 1

the curves $x^2 - 1$ and $2^x$

Online graph calculators show the two curves as touching.

If you zoom in sufficiently, their non-tangency becomes more apparent: the red and blue curves switch places.

values of their derivatives at $x=3$ are different. If two curves touch shouldn't their tangents have same slopes?

Indeed, two ‘touching’ curves have the same slope at their point of contact.

Answer 2

Here is the difference: $x^2-1-2^x$

Answer 3

Evaluate derivatives at $x=3\;$; they are not equal. They are not touching, ( although they seem to) but they intersect.

Somewhere else at about $x=2.2451$ the two curves have parallel tangents.