# Calculus Examples

Find the Tangent at a Given Point Using the Limit Definition
,
Step 1
The slope of the tangent line is the derivative of the expression.
The derivative of
Step 2
Consider the limit definition of the derivative.
Step 3
Find the components of the definition.
Evaluate the function at .
Replace the variable with in the expression.
Simplify the result.
Simplify each term.
Rewrite as .
Expand using the FOIL Method.
Apply the distributive property.
Apply the distributive property.
Apply the distributive property.
Simplify and combine like terms.
Simplify each term.
Multiply by .
Multiply by .
Reorder and .
Apply the distributive property.
Multiply by .
Apply the distributive property.
Reorder.
Move .
Move .
Reorder and .
Find the components of the definition.
Step 4
Plug in the components.
Step 5
Simplify.
Simplify the numerator.
Apply the distributive property.
Multiply by .
Multiply by .
Subtract from .
Subtract from .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Reduce the expression by cancelling the common factors.
Cancel the common factor of .
Cancel the common factor.
Divide by .
Reorder and .
Step 6
Evaluate the limit.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Evaluate the limit of which is constant as approaches .
Move the term outside of the limit because it is constant with respect to .
Evaluate the limit of which is constant as approaches .
Step 7
Evaluate the limit of by plugging in for .
Step 8
Multiply by .
Step 9
Simplify .
Multiply by .
Step 10
The slope is and the point is .
Step 11
Find the value of using the formula for the equation of a line.
Use the formula for the equation of a line to find .
Substitute the value of into the equation.
Substitute the value of into the equation.
Substitute the value of into the equation.
Find the value of .
Rewrite the equation as .
Multiply by .
Move all terms not containing to the right side of the equation.
Subtract from both sides of the equation.
Subtract from .
Step 12
Now that the values of (slope) and (y-intercept) are known, substitute them into to find the equation of the line.
Step 13