Calculus Examples

Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Since the exponent approaches , the quantity approaches .
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.
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Rewrite.
Differentiate using the Power Rule which states that is where .
Differentiate using the Exponential Rule which states that is where =.
Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Since the exponent approaches , the quantity approaches .
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.
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Rewrite.
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 .
Differentiate using the Exponential Rule which states that is where =.
Take the limit of each term.
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Move the term outside of the limit because it is constant with respect to .
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Multiply by .
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