Understanding L’Hospital’s Rule: Finding the Limit of sin(x)/x as x Approaches 0 and its Application

sinx/x (lim x -> 0)

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

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

L’Hospital’s Rule states that if we have an indeterminate form of the type 0/0 or ∞/∞, where a function f(x) and g(x) both approach 0 or infinity as x approaches a value, then the limit of the ratio of f(x) and g(x) can be found by taking the derivative of f(x) and g(x) separately and then evaluating the limit of the ratio of their derivatives.

Applying L’Hospital’s Rule to the given problem, we have:
lim (x -> 0) sin(x)/x

Taking the derivative of sin(x) with respect to x gives:
lim (x -> 0) cos(x)

Taking the derivative of x with respect to x gives:
lim (x -> 0) 1

Now we can evaluate the limit again:
lim (x -> 0) sin(x)/x = lim (x -> 0) cos(x)/1

As x approaches 0, the value of cos(x) approaches 1. Therefore, the final limit becomes:
lim (x -> 0) sin(x)/x = 1

So, the limit of sin(x)/x as x approaches 0 is equal to 1.

More Answers: