Take the limit of the numerator and the limit of the denominator.

Evaluate the limit of the numerator.

Take the limit of each term.

Split the limit using the Sum of Limits Rule on the limit as approaches .

Move the limit into the exponent.

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 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 by plugging in for .

Simplify the answer.

Simplify each term.

Multiply by .

Anything raised to is .

Multiply by .

Subtract from .

Add and .

Evaluate the limit of the denominator.

Move the exponent from outside the limit using the Limits Power Rule.

Evaluate the limit of by plugging in for .

Raising to any positive power yields .

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.

Differentiate the numerator and denominator.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

Differentiate using the Exponential Rule which states that is where =.

Replace all occurrences of with .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Multiply by .

Move to the left of .

Since is constant with respect to , the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Multiply by .

Add and .

Differentiate using the Power Rule which states that is where .

Move the term outside of the limit because it is constant with respect to .

Apply L’Hospital’s rule.

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.

Split the limit using the Sum of Limits Rule on the limit as approaches .

Move the term outside of the limit because it is constant with respect to .

Move the limit into the exponent.

Move the term outside of the limit because it is constant with respect to .

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 .

Simplify the answer.

Simplify each term.

Multiply by .

Anything raised to is .

Multiply by .

Subtract from .

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.

By the Sum Rule, the derivative of with respect to is .

Evaluate .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

Differentiate using the Exponential Rule which states that is where =.

Replace all occurrences of with .

Since is constant with respect to , the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Multiply by .

Move to the left of .

Multiply by .

Since is constant with respect to , the derivative of with respect to is .

Add and .

Differentiate using the Power Rule which states that is where .

Divide by .

Move the term outside of the limit because it is constant with respect to .

Move the limit into the exponent.

Move the term outside of the limit because it is constant with respect to .

Evaluate the limit of by plugging in for .

Combine and .

Multiply by .

Anything raised to is .

Multiply by .

Evaluate Using L’Hospital’s Rule limit as x approaches 0 of (e^(5x)-1-5x)/(x^2)