Use the same function f as above, but this time consider...

Use the same function $f$ as above, but this time consider what happens as $x$ approaches 1 . a. Study this situation experimentally as you did in parts a and $b$ of Problem 1. To gain some additional feel and respect for the situation, compute the numerator and denominator of $f$ separately for several $x$ values before dividing. What are your conclusions and, in particular, what is $\lim _{x \rightarrow 1} f(x)$ ? b. What happens when you try to "cheat" as was done in part $c$ of Problem 1? There are situations in which direct evaluation at the specified point is possible and actually gives the limit. These give rise to a concept called continuity. There are many important situations in calculus when this technique will not work, however.

Use the same function $f$ as above, but this time consider what happens as $x$ approaches 1 .
a. Study this situation experimentally as you did in parts a and $b$ of Problem 1. To gain some additional feel and respect for the situation, compute the numerator and denominator of $f$ separately for several $x$ values before dividing. What are your conclusions and, in particular, what is $\lim _{x \rightarrow 1} f(x)$ ?
b. What happens when you try to "cheat" as was done in part $c$ of Problem 1? There are situations in which direct evaluation at the specified point is possible and actually gives the limit. These give rise to a concept called continuity. There are many important situations in calculus when this technique will not work, however.

Key Concepts

Limits

A limit represents the value that a function approaches as the input nears a particular point, regardless of whether the function is defined at that point. This concept is fundamental in calculus as it provides a way to rigorously discuss the behavior of functions at points where they may not be explicitly defined or where direct evaluation is challenging.

Continuity

Continuity is the property of a function that allows the limit at a point to equal the function's value at that point. When a function is continuous, it means there is no interruption, jump, or hole at that point, thereby making techniques like direct substitution valid and efficient for evaluating limits.

Direct Substitution

Direct substitution is a straightforward method of evaluating a limit by plugging the value that x approaches directly into the function. This technique works seamlessly when the function is continuous at the point of interest, but it can fail in cases where the function exhibits indeterminate forms or discontinuities.

Experimental Evaluation of Functions

Experimental evaluation involves computing the values of a function at points near the value of interest to observe how the function behaves. This approach can provide empirical insight into the limiting behavior and help confirm analytic findings, especially in cases where direct methods are not easily applicable.

Transcript

00:01 Consider the shown function.

00:03 Okay, this is the function fx.

00:06 So let's see what will be the limit of fx when x tends to 1 from negative side.

00:19 That means x is taking the value of 1.

00:22 Sorry, x is taking the value 1 from negative side, this side.

00:28 So if you will just follow the trace the function, when the x, the x, becomes one the value of the function is approaching value of two so x when x tends to one from negative side value of the limit is two now let's see what will be the limit of the function limit of the function f x when x is approaching one from positive side now if you will trace the function from positive side okay trace the function from positive side the value of the function at equals to 1 is negative 1 right the value of the function here will be negative 1 you can see that negative 1 right so you can see the values are different from when x take the value from positive and negative side the values are different this one is called this one when it this is called sorry wait yeah this is called left -hand side limit and this is called right -hand limit now this these limits are different.

01:47 These limits are not equal.

01:51 The point is same, x equals to 1.

01:53 But when we approach this value from left hand side, the function approaches the value of 2.

01:59 And when we approach x equals to 1 from right hand side, the value of the function is minus negative 1.

02:05 So these limits are not equal.

02:07 That means limit of fx.

02:11 A particular x tends to 1 does not exist.

02:16 The limit does not exist here because left -hand side limit is different from right -hand -side limit.

02:26 So the left -hand -side limit is not equal to right -hand -side limit.

02:35 Therefore, the limit of the function at x -equals to 1 does not exist...