Exploring the Limit of sin(x)/x as x Approaches 0 Using L’Hospital’s Rule and the Limit Definition of the Derivative

lim x->0 sinx/x

To find the limit of sin(x)/x as x approaches 0, we can use the concept of L’Hospital’s Rule or the limit definition of the derivative

To find the limit of sin(x)/x as x approaches 0, we can use the concept of L’Hospital’s Rule or the limit definition of the derivative.

Using L’Hospital’s Rule:
Since both sin(x) and x approach 0 as x approaches 0, we can apply L’Hospital’s Rule which states that the limit of the ratio of the derivatives of two functions is equal to the limit of the ratio of the original functions.

Let’s differentiate sin(x) and x with respect to x:
d(sin(x))/dx = cos(x)
d(x)/dx = 1

Now we can take the limit of the ratio of the derivatives:
lim x->0 (cos(x))/1

As x approaches 0, cos(x) approaches 1. Therefore, the limit is equal to 1.

Using the limit definition of the derivative:
Another way to calculate the limit is by using the limit definition of the derivative.
The derivative of a function f(x) at a point x is defined as the limit:

f'(x) = lim h->0 (f(x+h) – f(x))/h

In this case, let’s consider f(x) = sin(x).
So, f'(x) = lim h->0 (sin(x+h) – sin(x))/h

Now, let’s calculate f'(x) for x = 0:
f'(0) = lim h->0 (sin(0+h) – sin(0))/h
= lim h->0 (sin(h))/h

As h approaches 0, we see that sin(h) also approaches 0.
Therefore, the limit simplifies to:

f'(0) = lim h->0 0/h = 0

So the limit of sin(x)/x as x approaches 0 is also equal to 0.

In conclusion, we can determine the limit of sin(x)/x as x approaches 0 to be either 1 or 0 depending on the method used. Both methods are valid and give us different insights into the behavior of the function as it approaches 0.

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