Calculus Examples

Evaluate the Limit limit as x approaches 0 of (sin(x))/(2x^2-x)
Step 1
Apply L'Hospital's rule.
Tap for more steps...
Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
Tap for more steps...
Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.1.2
Evaluate the limit of the numerator.
Tap for more steps...
Step 1.1.2.1
Move the limit inside the trig function because sine is continuous.
Step 1.1.2.2
Evaluate the limit of by plugging in for .
Step 1.1.2.3
The exact value of is .
Step 1.1.3
Evaluate the limit of the denominator.
Tap for more steps...
Step 1.1.3.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 1.1.3.2
Move the term outside of the limit because it is constant with respect to .
Step 1.1.3.3
Move the exponent from outside the limit using the Limits Power Rule.
Step 1.1.3.4
Evaluate the limits by plugging in for all occurrences of .
Tap for more steps...
Step 1.1.3.4.1
Evaluate the limit of by plugging in for .
Step 1.1.3.4.2
Evaluate the limit of by plugging in for .
Step 1.1.3.5
Simplify the answer.
Tap for more steps...
Step 1.1.3.5.1
Simplify each term.
Tap for more steps...
Step 1.1.3.5.1.1
Raising to any positive power yields .
Step 1.1.3.5.1.2
Multiply by .
Step 1.1.3.5.2
Add and .
Step 1.1.3.5.3
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.3.6
The expression contains a division by . The expression is undefined.
Undefined
Step 1.1.4
The expression contains a division by . The expression is undefined.
Undefined
Step 1.2
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.
Step 1.3
Find the derivative of the numerator and denominator.
Tap for more steps...
Step 1.3.1
Differentiate the numerator and denominator.
Step 1.3.2
The derivative of with respect to is .
Step 1.3.3
By the Sum Rule, the derivative of with respect to is .
Step 1.3.4
Evaluate .
Tap for more steps...
Step 1.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.4.2
Differentiate using the Power Rule which states that is where .
Step 1.3.4.3
Multiply by .
Step 1.3.5
Evaluate .
Tap for more steps...
Step 1.3.5.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.3.5.2
Differentiate using the Power Rule which states that is where .
Step 1.3.5.3
Multiply by .
Step 2
Evaluate the limit.
Tap for more steps...
Step 2.1
Split the limit using the Limits Quotient Rule on the limit as approaches .
Step 2.2
Move the limit inside the trig function because cosine is continuous.
Step 2.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 2.4
Move the term outside of the limit because it is constant with respect to .
Step 2.5
Evaluate the limit of which is constant as approaches .
Step 3
Evaluate the limits by plugging in for all occurrences of .
Tap for more steps...
Step 3.1
Evaluate the limit of by plugging in for .
Step 3.2
Evaluate the limit of by plugging in for .
Step 4
Simplify the answer.
Tap for more steps...
Step 4.1
The exact value of is .
Step 4.2
Simplify the denominator.
Tap for more steps...
Step 4.2.1
Multiply by .
Step 4.2.2
Multiply by .
Step 4.2.3
Subtract from .
Step 4.3
Divide by .