limit as x approaches 0: sinx/x
1
To evaluate the limit of sinx/x as x approaches 0, we can use L’Hopital’s rule or the Squeeze theorem.
Using L’Hopital’s rule:
lim x->0 sinx/x = lim x->0 (cosx/1) = cos(0)/1 = 1
Alternatively, we can use the Squeeze theorem. Since -1 ≤ sinx ≤ 1 for all values of x, we have:
-1/x ≤ sinx/x ≤ 1/x, as x > 0.
Taking the limits of both sides as x approaches 0 gives:
lim x->0 -1/x ≤ lim x->0 sinx/x ≤ lim x->0 1/x
As the limit on the right-hand side approaches infinity, and the limit on the left-hand side approaches negative infinity, the limit of sinx/x as x approaches 0 is equal to 1.
Therefore, the limit of sinx/x as x approaches 0 is equal to 1.
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