Limits Formula Sheet:
Trigonometric Limits:
sin 𝑥
𝑥
= lim
=1
𝑥→ 0
𝑥→ 0 sin 𝑥
𝑥
lim
tan 𝑥
𝑥
= lim
=1
𝑥→ 0
𝑥→ 0 tan 𝑥
𝑥
lim
1 − cos 𝑥
=0
𝑥→ 0
𝑥
lim
𝑠𝑖𝑛−1 (𝑥)
𝑥
= lim
=1
𝑥→ 0
𝑥→ 0 𝑠𝑖𝑛−1 (𝑥)
𝑥
lim
𝑡𝑎𝑛−1 (𝑥)
𝑥
lim
= lim
=1
𝑥→ 0
𝑥→ 0 𝑡𝑎𝑛−1 (𝑥)
𝑥
sin(𝑥 − 𝑎)
tan (𝑥 − 𝑎)
= lim
=1
𝑥→ 𝑎
𝑥→ 𝑎
𝑥−𝑎
𝑥−𝑎
lim
sin (1⁄𝑥 )
=1
𝑥→ ∞
1⁄𝑥
lim
sin 𝑥
cos 𝑥
= lim
=0
𝑥→ ∞
𝑥→ ∞
𝑥
𝑥
lim
Exponential Limits:
𝑥
lim 𝑒 = 1
𝑥→ 0
𝑒𝑥 − 1
lim
=1
𝑥→ 0
𝑥
1 𝑥
𝑥+1 𝑥
] = 𝑒
lim [1 + ] = lim [
𝑥→ ∞
𝑥→ ∞
𝑥
𝑥
lim [1 + 𝑥]1⁄𝑥 = 𝑒
𝑥→ 0
Logarithmic Limits:
ln (1 + 𝑥)
=1
𝑥→ 0
𝑥
lim
Other Limits:
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𝑎 𝑥
lim [1 + ] = 𝑒 𝑎
𝑥→ ∞
𝑥
𝑎𝑥 − 1
= ln 𝑎
𝑥→ 0
𝑥
lim
[1 + 𝑥 ] 𝑛 − 1
lim
=𝑛
𝑥→ 0
𝑥