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Continuity-HW Problems
1. For what values of 𝑥 are the following functions continuous?
2𝑥+1
𝑎. 𝑓(𝑥 ) = 𝑥 2 +4𝑥+3
𝑥 2 −9
𝑏. 𝑔(𝑥 ) = 𝑖𝑓 𝑥 ≠ −3
𝑥+3
=6 𝑖𝑓 𝑥 = −3
𝑐. ℎ(𝑥 ) = 𝑥 + 4 𝑖𝑓 𝑥>2
= 10 − 𝑥 2 𝑖𝑓 𝑥≤2
𝑑. 𝑓(𝑥 ) = √36 − 𝑥 2
3
𝑒. 𝑔(𝑥 ) = √36 − 𝑥 2
2. Evaluate the following limits:
cos2 𝑥−1
𝑎. lim √
𝑥→0 cos 𝑥−1
𝑥 2 −1
𝑏. lim sin( 𝑥−1 )
𝑥→1
3. Suppose 𝑓 (𝑥 ) = √𝑥 2 + 4 + 𝑥 − 3. Show that 𝑓(𝑥) has a point 𝑐,
where 0 < 𝑐 < √21 such that 𝑓(𝑐 ) = 0.
4. Suppose 𝑓 (𝑥 ) = 𝑐𝑜𝑠𝑥 − 𝑥 5 . Show that 𝑓 (𝑥 ) has a point 𝑐, where
0 < 𝑐 < 2 such that 𝑓(𝑐 ) = 0.
Continuity-HW Problems
1. For what values of 𝑥 are the following functions continuous?
2𝑥+1
𝑎. 𝑓(𝑥 ) = 𝑥 2 +4𝑥+3
𝑥 2 −9
𝑏. 𝑔(𝑥 ) = 𝑖𝑓 𝑥 ≠ −3
𝑥+3
=6 𝑖𝑓 𝑥 = −3
𝑐. ℎ(𝑥 ) = 𝑥 + 4 𝑖𝑓 𝑥>2
= 10 − 𝑥 2 𝑖𝑓 𝑥≤2
𝑑. 𝑓(𝑥 ) = √36 − 𝑥 2
3
𝑒. 𝑔(𝑥 ) = √36 − 𝑥 2
2. Evaluate the following limits:
cos2 𝑥−1
𝑎. lim √
𝑥→0 cos 𝑥−1
𝑥 2 −1
𝑏. lim sin( 𝑥−1 )
𝑥→1
3. Suppose 𝑓 (𝑥 ) = √𝑥 2 + 4 + 𝑥 − 3. Show that 𝑓(𝑥) has a point 𝑐,
where 0 < 𝑐 < √21 such that 𝑓(𝑐 ) = 0.
4. Suppose 𝑓 (𝑥 ) = 𝑐𝑜𝑠𝑥 − 𝑥 5 . Show that 𝑓 (𝑥 ) has a point 𝑐, where
0 < 𝑐 < 2 such that 𝑓(𝑐 ) = 0.
