Limit of (1 – cos(x)) / x as x approaches 0 | Discover the key manipulations and applications of L’Hopital’s rule

limit as x approaches 0: 1-cosx/x

To find the limit of the function as x approaches 0, we can use direct substitution first and see if it yields a determinate form

To find the limit of the function as x approaches 0, we can use direct substitution first and see if it yields a determinate form.

Let’s substitute x=0 into the function:

1 – cos(0) / 0 = 1 – 1 / 0

We know that division by zero is undefined, so direct substitution does not work in this case.

However, we can manipulate the expression algebraically to find an equivalent expression that is easier to evaluate.

Using the fact that cos(0) = 1 and cos^2(x) + sin^2(x) = 1, we can rewrite the expression as:

(1 – cos(x)) / x * (1 + cos(x)) / (1 + cos(x))

Simplifying this expression, we get:

(1 – cos^2(x)) / (x * (1 + cos(x)))

Now, we can see that the numerator can be factored:

(1 – cos(x)) (1 + cos(x)) / (x * (1 + cos(x)))

The (1 + cos(x)) terms cancel out:

(1 – cos(x)) / x

Now we can substitute x=0:

(1 – cos(0)) / 0 = (1 – 1) / 0 = 0 / 0

We have reached another indeterminate form, 0/0, which is still undefined.

To further evaluate the limit, we can apply L’Hopital’s rule. This rule allows us to differentiate the numerator and denominator and then re-evaluate the limit.

Differentiating the numerator and denominator with respect to x:

d/dx (1 – cos(x)) / d/dx (x) = sin(x) / 1 = sin(x)

Now we can substitute x=0 into the derivative:

sin(0) = 0

Therefore, the limit as x approaches 0 of (1 – cos(x)) / x is 0.

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