Question

If a 4-m-tall child stands 2 m in front of a vertical plane mirror, the image of the child will be ------- m tall
1
2
3
4

Answer 1

Answer on Question #46995 – Math – Calculus

Question. Discuss the continuity of the function $f$, defined by

$$f(x) = \begin{cases} x^2 - 1, & x \leq 1, \\ 1 - 1/x, & x \geq 1. \end{cases}$$

Solution. 1) Suppose $x < 1$. Then $f(x) = x^2 - 1$ is a polynomial, and therefore it is continuous at each such $x$.

2) Suppose $x = 1$. Then left limit of $f$ at $x = 1$ is equal to

$$\lim_{x \to 1 - 0} f(x) = \lim_{x \to 1 - 0} x^2 - 1 = 1^2 - 1 = 1 - 1 = 0,$$

and the right limit is

$$\lim_{x \to 1 + 0} f(x) = \lim_{x \to 1 + 0} 1 - 1/x = 1 - 1/1 = 1 - 1 = 0.$$

Thus left and right limits of $f$ at $x = 1$ coincide, and therefore $f$ is continuous at $x = 1$.

3) Finally, let $x > 1$. Then $f(x) = 1 - 1/x$. Since $x \neq 0$, this function is continuous at all such $x$.

Thus $f$ is continuous at all $x \in \mathbb{R}$.

Answer. $f$ is continuous at all $x \in \mathbb{R}$.

http://www.AssignmentExpert.com/