Long Test in Calculus - Limits of Trigonometric Functions
I. Evaluate the following limits. Show your complete solution and box
your final answer. No solution, no credits. (21 points)
1. lim
𝑡𝑎𝑛 𝑥
𝑥
𝑥→0
2. lim
1 − 𝑐𝑜𝑠 2𝑥
3𝑥
𝑥→0
3. lim [
1
𝑥→0 𝑐𝑜𝑠 𝑥
4. lim
𝑥→0
𝑠𝑖𝑛 4𝑥
𝑥
+
𝑡𝑎𝑛 𝑥
𝑥
]
5. lim 𝑠𝑖𝑛
𝑥→𝜋
6. lim
𝑥
3
𝑠𝑖𝑛 𝑥
𝑥→0 𝑐𝑜𝑠 𝑥
7. lim [𝑐𝑜𝑠
𝑥→𝜋
5𝑥
6
𝑥
− 𝑐𝑜𝑠 ]
6
II. If k is any real number, can you prove that lim
𝑥→0
lim
𝑥→0
1 − 𝑐𝑜𝑠 𝑘𝑥
𝑘𝑥
𝑠𝑖𝑛 𝑘𝑥
𝑘𝑥
= 1 and
= 0? Explain in not more than 5 sentences. You may also
show sample solution/s. (4 points)