Calculus Examples
Step 1
Find where the expression is undefined.
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 2
The vertical asymptotes occur at areas of infinite discontinuity.
No Vertical Asymptotes
Step 3
Step 3.1
Apply L'Hospital's rule.
Step 3.1.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 3.1.1.1
Take the limit of the numerator and the limit of the denominator.
Step 3.1.1.2
Evaluate the limit of the numerator.
Step 3.1.1.2.1
Evaluate the limit.
Step 3.1.1.2.1.1
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 3.1.1.2.1.2
Evaluate the limit of which is constant as approaches .
Step 3.1.1.2.2
Since the function approaches , the positive constant times the function also approaches .
Step 3.1.1.2.2.1
Consider the limit with the constant multiple removed.
Step 3.1.1.2.2.2
Since the exponent approaches , the quantity approaches .
Step 3.1.1.2.3
Infinity plus or minus a number is infinity.
Step 3.1.1.3
Since the exponent approaches , the quantity approaches .
Step 3.1.1.4
Infinity divided by infinity is undefined.
Undefined
Step 3.1.2
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 3.1.3
Find the derivative of the numerator and denominator.
Step 3.1.3.1
Differentiate the numerator and denominator.
Step 3.1.3.2
By the Sum Rule, the derivative of with respect to is .
Step 3.1.3.3
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3.4
Evaluate .
Step 3.1.3.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3.4.2
Differentiate using the Exponential Rule which states that is where =.
Step 3.1.3.5
Add and .
Step 3.1.3.6
Differentiate using the Exponential Rule which states that is where =.
Step 3.1.4
Reduce.
Step 3.1.4.1
Cancel the common factor of .
Step 3.1.4.1.1
Cancel the common factor.
Step 3.1.4.1.2
Rewrite the expression.
Step 3.1.4.2
Cancel the common factor of .
Step 3.1.4.2.1
Cancel the common factor.
Step 3.1.4.2.2
Divide by .
Step 3.2
Evaluate the limit of which is constant as approaches .
Step 4
List the horizontal asymptotes:
Step 5
There is no oblique asymptote because the degree of the numerator is less than or equal to the degree of the denominator.
No Oblique Asymptotes
Step 6
This is the set of all asymptotes.
No Vertical Asymptotes
Horizontal Asymptotes:
No Oblique Asymptotes
Step 7