Step 1: The Indeterminate Form
Evaluate . Notice it evaluates to .
Action: Slider to Zoom In to the origin to see what happens to the curves!
At high zoom, the curves look like their tangent lines!
What is the ratio of their slopes (Red Slope / Blue Slope)?
Step 2: Apply L'Hopital's Rule
Since the limit is , we can take the derivative of the top and bottom separately.
Action: Drag the d/dx operator to both parts of the fraction.
Derivatives applied! Now plug in x = {{ s2Prob.limit }}:
Step 3: Tangent Approximation
To estimate values near a known point, we use the tangent line!
Action: Adjust the slope to make the line perfectly tangent to at .
Step 4: Build the Formula
Estimate .
Action: Tap the bubbles below to fill in the linear approximation formula:
Equation Complete!
Mastery Achieved!
You've successfully evaluated limits with L'Hopital's Rule and estimated values using Linear Approximation!