limit as x approaches 0: 1-cosx/x
To find the limit as x approaches 0 for the function 1 – cos(x) / x, we can simplify and evaluate it directly
To find the limit as x approaches 0 for the function 1 – cos(x) / x, we can simplify and evaluate it directly.
Let’s simplify the expression first:
1 – cos(x) / x = (x – cos(x)) / x
Now, as x approaches 0, we have to consider that cos(x) approaches 1. Therefore, the numerator (x – cos(x)) approaches 0, and the denominator (x) also approaches 0.
To evaluate the limit, we can apply L’Hospital’s Rule, which states that if we have an indeterminate form of the type 0/0, taking the derivative of the numerator and denominator separately may help in evaluating the limit.
Taking the derivative of the numerator and denominator, we get:
d/dx (x – cos(x)) = 1 + sin(x)
d/dx (x) = 1
Now, let’s rewrite the function after taking the derivatives:
(1 + sin(x)) / 1
As x approaches 0, sin(x) approaches 0 as well. Plugging in x = 0 into the function, we have:
(1 + sin(0)) / 1 = (1 + 0) / 1 = 1 / 1 = 1
Thus, the limit as x approaches 0 for the function 1 – cos(x) / x is equal to 1.
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