Question

Calculus limit problem

(a) Explain why the following calculation is incorrect.

lim ((1/x) - (1/x^2)) = lim 1/x - lim 1/x^2
x --> 0+ x->0+ x->0+

= +infinity - (+infinity) = 0


(b) Show that lim ((1/x) - (1/x^2)) = -infinity
x->0+

Thanks for your help!

Answer 1

Like everything in math, those addition/subtraction limit properties have hypothesis requirements. You can apply them only when your limits meet the requirements

Answer 2

\[\lim\limits_{x\to a} \left[f(x) - g(x)\right] =\lim\limits_{x\to a} f(x) - \lim\limits_{x\to a} g(x) \]
is true only when both the individual limits exist.

Answer 3

When your limit is \(\lim_{x\to0^+} \frac{1}{x}\) or \(\lim_{x\to0^+} \frac{1}{x^2}\) approaching (positive) x to 0 does get you closer and closer to infinity.
However, what you're asked to find out isn't any of those limits, but limit of the their difference.
Which means, what does the difference of those expressions get closer and closer to as x approaches 0.

Saying the limit approaches 0 means the expressions get closer and closer to each other as x approaches 0, but as you can see:
$$\lim_{x\to0^+} \bigg[\frac{1}{x} - \frac{1}{x^2}\bigg] = \lim_{x\to0^+} \bigg[\frac{1}{x} - \bigg(\frac{1}{x}\bigg)^2\bigg]$$
And clearly, when \(\frac{1}{x}\) gets bigger, \(\big(\frac{1}{x} \big)^2\) gets even bigger, so they get farther and farther from each other and therefore the difference between them gets bigger and bigger, so it's pretty intuitive to see why when doing the \((small-big)\) you get \((-\infty)\).

In order to evaluate the difference between the expressions we can factor the expression:
$$
\lim_{x\to0^+} \bigg[\frac{1}{x} - \frac{1}{x^2}\bigg] = \lim_{x\to0^+} \bigg[\frac{1}{x} \cdot \bigg(1 - \frac{1}{x}\bigg) \bigg]
$$We have a single expression now: \(\lim_{x\to0^+} \frac{1}{x} = \infty\).
As x approaches 0 this expression approaches infinity.
We can see how it affects the limit:
$$
\bigg(\lim_{x\to0^+} \frac{1}{x} \bigg) \cdot\bigg[1 - \bigg(\lim_{x\to0^+}\frac{1}{x}\bigg) \bigg] = \infty \cdot (1 - \infty) = \infty \cdot (-\infty) = -\infty^2 = -\infty
$$

Answer 4

@ganeshie8 The limits do exist.

The problem is that \(+\infty - +\infty\) is an indeterminate form, just like \(0/0\), so you can't just assume they cancel out to \(0\).