TWO examples of two sided limits and
each theorem on limits.
Sure! Here are two examples of two-sided limits:
1. Example 1: Find the limit of f(x) = (4x^2 - 1)/(x - 1) as x approaches 1.
To find the limit, we can directly substitute the value of x into the function. However, in this case, substituting x = 1 gives us an indeterminate form (0/0). So, we need to use an algebraic technique called factorization. By factoring the numerator using the difference of squares, we get f(x) = (2x + 1)(2x - 1)/(x - 1). Now, we can cancel out the (x - 1) term, resulting in f(x) = (2x + 1)(2x - 1). Then, substituting x = 1 into the simplified function, we find that the limit is 3.
2. Example 2: Find the limit of g(x) = sin(x)/x as x approaches 0.
To evaluate this limit, we can first try directly substituting x = 0 into the function. However, this results in the indeterminate form (0/0). In this case, we can use a trigonometric limit theorem called the Squeeze Theorem. The Squeeze Theorem states that if f(x) ≤ g(x) ≤ h(x) near a certain point, and the limits of f(x) and h(x) as x approaches that point are both equal to L, then the limit of g(x) as x approaches that point is also L. By using this theorem, we know that -1 ≤ sin(x)/x ≤ 1 for all x ≠ 0. Thus, the limit of g(x) as x approaches 0 is also equal to 1.
Now, here are a couple of theorems on limits:
1. The Squeeze Theorem: As mentioned above, the Squeeze Theorem is used to evaluate the limit of a function if it is bounded between two other functions that have the same limiting value.
2. The Limit Laws: The Limit Laws consist of several rules that help simplify the evaluation of limits. These laws include properties such as the sum, difference, product, and quotient of limits, the power rule, the exponential and logarithmic limit rules, and the trigonometric limit rules. These laws allow us to manipulate functions and find their limits more easily.
It's important to note that there are many more theorems and techniques to evaluate limits depending on the specific mathematical context.