What is the value of the limit lim(x→0) (sin(5x)/x)?
Practice Questions
Q1
What is the value of the limit lim(x→0) (sin(5x)/x)?
0
1
5
10
Questions & Step-by-Step Solutions
What is the value of the limit lim(x→0) (sin(5x)/x)?
Step 1: Identify the limit we want to evaluate: lim(x→0) (sin(5x)/x).
Step 2: Notice that if we directly substitute x = 0, we get sin(0)/0, which is 0/0. This is an indeterminate form.
Step 3: Since we have an indeterminate form, we can use L'Hôpital's rule. This rule states that if we have 0/0 or ∞/∞, we can take the derivative of the numerator and the derivative of the denominator.
Step 4: Differentiate the numerator: The derivative of sin(5x) is 5cos(5x).
Step 5: Differentiate the denominator: The derivative of x is 1.
Step 6: Now we rewrite the limit using the derivatives: lim(x→0) (5cos(5x)/1).
Step 7: Substitute x = 0 into the new limit: 5cos(5*0) = 5cos(0) = 5*1 = 5.
