# Evaluate ( limit as x approaches 0 of sin(ax))/(sin(bx))

Multiply the numerator and denominator by .
Multiply the numerator and denominator by .
Separate fractions.
Split the limit using the Product of Limits Rule on the limit as approaches .
Take the limit of each term.
Move the term outside of the limit because it is constant with respect to .
Move the term outside of the limit because it is constant with respect to .
Move the term outside of the limit because it is constant with respect to .
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Take the limit of each term.
Move the limit inside the trig function because sine is continuous.
Move the term outside of the limit because it is constant with respect to .
Evaluate the limit of by plugging in for .
Multiply by .
The exact value of is .
Evaluate the limit of by plugging in for .
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Remove parentheses.
Differentiate using the Power Rule which states that is where .
Multiply by .
Multiply by .
Differentiate using the Power Rule which states that is where .
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of by plugging in for .
Evaluate the limit of the denominator.
Take the limit of each term.
Move the limit inside the trig function because sine is continuous.
Move the term outside of the limit because it is constant with respect to .
Evaluate the limit of by plugging in for .
Multiply by .
The exact value of is .
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
Differentiate using the Power Rule which states that is where .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
The derivative of with respect to is .
Replace all occurrences of with .
Since is constant with respect to , the derivative of with respect to is .
Remove parentheses.
Differentiate using the Power Rule which states that is where .
Multiply by .
Multiply by .
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of by plugging in for .
Evaluate the limit of by plugging in for .
The expression contains a division by The expression is undefined.
Undefined
Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
Differentiate using the Power Rule which states that is where .
Differentiate using the Power Rule which states that is where .
Take the limit of each term.
Split the limit using the Limits Quotient Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Move the limit inside the trig function because cosine is continuous.
Move the term outside of the limit because it is constant with respect to .
Split the limit using the Limits Quotient Rule on the limit as approaches .
Move the term outside of the limit because it is constant with respect to .
Move the limit inside the trig function because cosine is continuous.
Move the term outside of the limit because it is constant with respect to .
Split the limit using the Limits Quotient Rule on the limit as approaches .
Evaluate the limits by plugging in for all occurrences of .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply by .
Multiply and .
Multiply by .
Multiply and .
Combine and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply by .
Evaluate ( limit as x approaches 0 of sin(ax))/(sin(bx))