Q. 13 - Calculus

Question

Describe in terms of large and small numbers why it makes intuitive sense that limits of the form (a) $\frac{1}{0^{+}}$ (b) $\frac{\infty }{0^{+}}$(c) $\frac{\infty }{1}$ must be infinite.

Step-by-Step Solution

Answer

(a) The limit $\frac{1}{0^{+}}$ is infinity and is very large.

(b) The limit $\frac{\infty }{0^{+}}$ is infinity and is very large.

(c) The limit $\frac{\infty }{1}$ is infinite.

Step 1. Given Information

The given limits are $\frac{\infty }{0^{+}},\frac{\infty }{0^{+}}$and $\frac{\infty }{1}$

Part (a) Step 1. Explanation of Limit of the form 1 0 +

$\underset{x\to 0^{+}}{\lim }\frac{1}{x}=\infty$

The limit is approaching from right and any number divide by zero is infinity.

 A number when divided by zero gives infinity.

Part (b) Step 1. Explanation of limit ∞ 0 +

$\underset{x\to 0^{+}}{\lim }\frac{1}{x}=\infty$

The limit is approaching from right and any number divide by zero is infinity.

Part (c) Step 1. Explanation of limit ∞ 1

The limit $\frac{\infty }{1}$is $\infty$ because the numerator has $\infty$ and any number when divided by 1, result itself.