Understanding the Notation lim x→1- and Evaluating Limits from the Left Side

Lim x–>1-

The notation “lim x→1-” represents the limit as x approaches 1 from the left side

The notation “lim x→1-” represents the limit as x approaches 1 from the left side. This means that we consider the values of x that are slightly smaller than 1. To evaluate this limit, we need to examine the behavior of the function as x approaches 1 from the left.

Let’s consider a function f(x) and assume that x approaches 1 from values smaller than 1. To evaluate lim x→1-, we can follow these steps:

1. Substitute x with values slightly smaller than 1 in the function f(x).
2. Observe the behavior of f(x) as those values of x approach 1.
3. Determine if f(x) approaches a specific value or if it approaches infinity or negative infinity as x gets closer to 1 from the left side.

Here’s an example to illustrate the concept:

Consider the function f(x) = (x + 1)/(x – 1).

Let’s substitute values slightly smaller than 1 for x and observe the behavior of the function:

When x = 0.9, f(0.9) = (0.9 + 1)/(0.9 – 1) = 1.9/-0.1 = -19
When x = 0.99, f(0.99) = (0.99 + 1)/(0.99 – 1) = 1.99/-0.01 = -199
When x = 0.999, f(0.999) = (0.999 + 1)/(0.999 – 1) = 1.999/-0.001 = -1999

As x approaches 1 from the left side, we can see that the values of f(x) become more negative without bound (tending towards negative infinity). Therefore, the limit as x approaches 1 from the left side of the function f(x) is negative infinity.

Please note that this is just an example to illustrate the process of evaluating a limit as x approaches 1 from the left side. The specific function and its behavior may vary depending on the context of the problem.

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