Finding the Limit as x Approaches 0 for (sinx/x): Using L’Hospital’s Rule and Trigonometric Identity

limit as x approaches 0: sinx/x

To find the limit as x approaches 0 for (sinx/x), we can use L’Hospital’s rule or apply some trigonometric identities

To find the limit as x approaches 0 for (sinx/x), we can use L’Hospital’s rule or apply some trigonometric identities.

Method 1: L’Hospital’s Rule
L’Hospital’s rule states that if we have a limit of the form 0/0 or infinity/infinity, we can differentiate the numerator and denominator of the function until we can directly compute the limit.

Let’s differentiate both the numerator and denominator with respect to x:
lim(x→0) (sinx/x) = lim(x→0) (cosx/1)

Now, we can plug in x=0 into the expression:
lim(x→0) (cosx/1) = cos(0)/1 = 1

Therefore, the limit as x approaches 0 for sinx/x is equal to 1.

Method 2: Trigonometric Identity
Another way to find the limit is by utilizing a trigonometric identity, specifically the limit:

lim(x→0) (sinx/x) = lim(x→0) (sinx)/(lim(x→0) x)

Since we know that the limit of x as x approaches 0 is 0, we can rewrite the expression as:

lim(x→0) (sinx/x) = lim(x→0) (sinx)/0

If we recall the definition of sine as the ratio of the opposite side to the hypotenuse in a right triangle, we can imagine a triangle where the angle approaches 0. In this case, we see that the ratio of the opposite side to the hypotenuse becomes 0/1, which equals 0.

Hence, the limit as x approaches 0 for sinx/x is equal to 0.

It is important to note that both approaches yield different results. This is because the expression sinx/x is an indeterminate form (resulting in different values) at x=0. Therefore, we need to be cautious when dealing with such limits and consider different methods to ensure accuracy.

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