one-sided limits
One-sided limits are a concept in calculus that help us understand the behavior of a function as the input approaches a specific value, either from the left side (approaching from values less than the given value) or from the right side (approaching from values greater than the given value)
One-sided limits are a concept in calculus that help us understand the behavior of a function as the input approaches a specific value, either from the left side (approaching from values less than the given value) or from the right side (approaching from values greater than the given value). One-sided limits are denoted as follows:
For the left-hand limit: lim(x → a-) f(x)
For the right-hand limit: lim(x → a+) f(x)
Let’s explain this further with examples:
Example 1:
Consider the function f(x) = x^2. What happens to the function as x approaches 2 from the left side? We can write this as lim(x → 2-) x^2.
To find the left-hand limit, we evaluate the function for x values that are less than 2. For example, if we take x = 1.9, f(1.9) = (1.9)^2 = 3.61. As we continue to take x values closer to 2 from the left side, we see that f(x) gets closer to 4.
Therefore, lim(x → 2-) x^2 = 4.
The left-hand limit tells us how the function behaves as x approaches 2 from the left side.
Example 2:
Consider the function g(x) = 1/x. What happens to the function as x approaches 0 from the right side? We can write this as lim(x → 0+) 1/x.
To find the right-hand limit, we evaluate the function for x values that are greater than 0. For example, if we take x = 0.1, g(0.1) = 1/(0.1) = 10. As we continue to take x values closer to 0 from the right side, we see that g(x) approaches infinity (∞) because the denominator becomes infinitesimally close to 0.
Therefore, lim(x → 0+) 1/x = ∞.
The right-hand limit tells us how the function behaves as x approaches 0 from the right side.
In summary, one-sided limits allow us to analyze the behavior of functions as the input approaches a specific value from either the left or right side. It helps us understand the existence of limits, determine if they are equal from both sides, and establish whether a function is continuous at a given point.
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