Step 1: Identify the limit we want to find: lim(x→0) (sin(5x)/x).
Step 2: Substitute x = 0 into the expression. We get sin(5*0)/0 = sin(0)/0, which is 0/0. This is an indeterminate form.
Step 3: Since we have an indeterminate form (0/0), we can use L'Hôpital's rule. This rule states that 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 expression: 5cos(5*0) = 5cos(0).
Step 8: We know that cos(0) = 1, so we have 5 * 1 = 5.
Step 9: Therefore, the limit lim(x→0) (sin(5x)/x) = 5.
Limit of a Function – Understanding how to evaluate the limit of a function as it approaches a specific value, particularly using L'Hôpital's rule for indeterminate forms.
L'Hôpital's Rule – Application of L'Hôpital's rule to resolve limits that result in indeterminate forms like 0/0.
Trigonometric Limits – Knowledge of the behavior of trigonometric functions near zero, specifically that sin(x)/x approaches 1 as x approaches 0.
