Andrew Wright said:
So my working would be something like
lim (2x+5/x) x-> inf = 2 |Z+| +5 / |Z+| = 2 |Z+| / |Z+| = 2
The expression in your limit above is ##2x + \frac 5 x##, which undoubtedly isn't what you meant. If I take what you wrote literally, I get this:
$$\lim_{x \to \infty} 2x + \frac 5 x = \lim_{x \to \infty} 2x + \lim_{x \to \infty} \frac 5 x = \infty + 0 = \infty$$
Although you used parentheses in what you wrote, they are missing where they are needed the most - in the numerator. As linear text, the expression should be (2x + 5)/x.
There is another problem with what you wrote; namely the expression 2 |Z+| /|Z+|, which is ##\frac \infty \infty##, which isn't defined.
The limit you probably intended is this:
$$ \lim_{x \to \infty} \frac {2x + 5}x = \lim_{x \to \infty} \frac {2x}x +\lim_{x \to \infty} \frac 5 x$$
Using the properties of limits at infinity, the above simplifies to 2 + 0 = 2.
The more rigorous approach alluded to in this thread goes something like this:
For any (very large) number M, there exists a positive number ##\epsilon## near 0 such that if x > M, then ##\frac {2x + 5}x - 2 < \epsilon##.
WWGD said:
A metric on a space S is a function #d: S \ times S \right arrow \mathbb R # so d(x,y) must be a Real number.
For TeX, use
two # signs at each end (inlne) or
two $ signs at each end (standalone).
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