L'Hopital Practice Problems
Let's practice L'Hopital's Rule with a variety of problems. The key is recognizing when it applies and executing it correctly.
Exponential Limit
Hard
Find lim[x->0] (e^x - 1 - x)/x^2
1
Check form
(e^0 - 1 - 0)/0^2 = (1-1-0)/0 = 0/0 ✓
2
L'Hopital #1
= lim (e^x - 1)/(2x)
At x=0: (1-1)/0 = 0/0 ✓
3
L'Hopital #2
= lim e^x/2 = e^0/2 = 1/2
Answer:
1/2
Logarithmic Limit
Medium
Find lim[x->1] ln(x)/(x - 1)
1
Check form
ln(1)/(1-1) = 0/0 ✓
2
L'Hopital
= lim (1/x)/1 = 1/x at x=1 = 1
Answer:
1
Comprehensive Practice
1. lim[x->0] tan(x)/x = ?
0/0 form. d/dx[tan(x)] = sec^2(x). So lim sec^2(x)/1 = sec^2(0) = 1
2. lim[x->0] (e^(2x) - 1)/x = ?
0/0 form. L'Hopital: lim 2e^(2x)/1 = 2e^0 = 2
3. lim[x->infinity] x/e^x = ?
infinity/infinity. L'Hopital: lim 1/e^x = 0 as x->infinity
Key Takeaways
L'Hopital works great for exponential, logarithmic, and trig limits
Always verify you have 0/0 or infinity/infinity before applying
Keep applying until you get a determinate form
Exponentials grow faster than polynomials: x^n/e^x -> 0