Chapter 13.1, Problem 1E

When we write $\underset{x\to a}{\lim }f(x)=L$ then, roughly speaking, the values of f(x) get closer and closer to the number _______ as the values of x get closer and closer to _______. To determine $\underset{x\to 5}{\lim }\frac{x-5}{x-5}$, we try values for x closer and closer to _______ and find that the limit is _______.

To determine

To fill: When we write $\underset{x\to a}{\lim }f(x)=L$ then, roughly speaking the values of $f(x)$ get closer and closer to the number ______ as the values of $x$ get closer and closer to_____.

To determine $\underset{x\to 5}{\lim }\frac{x-5}{x-5}$, we try values of $x$ closer and closer to ______ and find that the limit is ______.

Answer to Problem 1E

When we write $\underset{x\to a}{\lim }f(x)=L$ then, roughly speaking the values of $f(x)$ get closer and closer to the number $\underset{\_}{L}$ as the values of $x$ get closer and closer to $\underset{\_}{a}$.

To determine $\underset{x\to 5}{\lim }\frac{x-5}{x-5}$, we try values of $x$ closer and closer to $\underset{\_}{5}$ and find that the limit is $\underset{\_}{1}$.

Explanation of Solution

We write $\underset{x\to a}{\lim }f(x)=L$ and say that the “left hand limit of $f(x)$ as $x$ approaches a” is equal to L.

If we make values of $f(x)$ arbitrarily close to $L$ by make $x$ sufficiently close to a and x less than a.

This definition varies from definition of two sided limit as in that we require x to be less than a.

In similar fashion if we want $x$ greater than $a$ we get right hand limit of $f(x)$ as $x$ approaches a.

Thus, $\underset{x\to a}{\lim }f(x)=L$ roughly means that $f(x)$ get closer and closer to the number $L$ as the values of $x$ get closer and closer to $a$.