The Limit of (1 – cosx)/x as x Approaches 0: Simplification and Evaluation

(1-cosx)/x (lim x -> 0)

To simplify this expression, we can start by using the identity:

lim x->0 (1 – cosx)/x = lim x->0 (1 – cosx)/(x-0)

Now, we can use the limit definition to evaluate this expression:

lim x->0 (1 – cosx)/(x-0) = lim x->0 (1 – cosx)/x

Since we have a 0/0 indeterminate form, we can use L’Hopital’s rule to further simplify the expression:

lim x->0 (1 – cosx)/x = lim x->0 (-sinx)/1

Now, we can evaluate the limit:

lim x->0 (-sinx)/1 = -sin(0)/1 = 0/1 = 0

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

To simplify this expression, we can start by using the identity:

lim x->0 (1 – cosx)/x = lim x->0 (1 – cosx)/(x-0)

Now, we can use the limit definition to evaluate this expression:

lim x->0 (1 – cosx)/(x-0) = lim x->0 (1 – cosx)/x

Since we have a 0/0 indeterminate form, we can use L’Hopital’s rule to further simplify the expression:

lim x->0 (1 – cosx)/x = lim x->0 (-sinx)/1

Now, we can evaluate the limit:

lim x->0 (-sinx)/1 = -sin(0)/1 = 0/1 = 0

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

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