Finding the Limit of f(x) = 1/x^2 as x Approaches 0 | Analysis and Explanation

lim x->0 (1/x^2)

To find the limit of a function as x approaches a certain value, we need to evaluate the behavior of the function as x gets arbitrarily close to that value

To find the limit of a function as x approaches a certain value, we need to evaluate the behavior of the function as x gets arbitrarily close to that value. In this case, we want to find the limit as x approaches 0 for the function f(x) = 1/x^2.

Let’s begin by substituting values closer and closer to 0 and observe the corresponding function values:

x = 0.1: f(0.1) = 1/(0.1)^2 = 100
x = 0.01: f(0.01) = 1/(0.01)^2 = 10,000
x = 0.001: f(0.001) = 1/(0.001)^2 = 1,000,000

As you can see, as x approaches 0, the function values become increasingly larger. In fact, for any positive value of x, f(x) will always be positive. On the other hand, if x approaches 0 from the negative side, the function will still be positive but the values will be the same since squaring a negative number gives a positive result. Therefore, we can conclude that the function does not approach a finite value as x approaches 0.

Instead, as x gets closer and closer to 0, the function diverges towards positive infinity. Mathematically, we can write this as:

lim x->0 (1/x^2) = ∞ or lim x->0 (1/x^2) = +∞

This means that the function 1/x^2 becomes arbitrarily large (tends to infinity) as x approaches 0.

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