Limits Formula Sheet

Basic Limits involving sine, cosine, and tangent:
- \lim_{x \to 0} \frac{\sin x}{x} = 1
- \lim_{x \to 0} \frac{x \sin x}{x} = 1
- \lim_{x \to 0} \frac{\tan x}{x} = 1
- \lim_{x \to 0} \frac{x \tan x}{x} = 1
- \lim_{x \to 0} \frac{1 - \cos x}{x} = 0
Inverse Sine and Tangent Limits:
- \lim_{x \to 0} \frac{\sin^{-1}(x)}{x} = 1
- \lim_{x \to 0} \frac{x}{\sin^{-1}(x)} = 1
- \lim_{x \to 0} \frac{\tan^{-1}(x)}{x} = 1
- \lim_{x \to 0} \frac{x}{\tan^{-1}(x)} = 1
Limits involving constants:
- \lim_{x \to a} \frac{\sin(x - a)}{x - a} = 1
- \lim_{x \to a} \frac{\tan(x - a)}{x - a} = 1
Limits as x approaches infinity for sine and cosine:
- \lim_{x \to \infty} \frac{\sin(\frac{1}{x})}{\frac{1}{x}} = 1
- \lim_{x \to \infty} \frac{\sin x}{x} = 0
- \lim_{x \to \infty} \frac{\cos x}{x} = 0

Basic Limits relating to the exponential function:
- \lim_{x \to 0} e^x = 1
- \lim_{x \to 0} \frac{e^x - 1}{x} = 1
- \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e
- \lim_{x \to \infty} \left(x + \frac{1}{x}\right)^x = e
Exponential limits as x approaches 0:
- \lim_{x \to 0} \left(1 + x\right)^{\frac{1}{x}} = e
Limits involving constants:
- \lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a

Basic Limits involving natural logarithm:
- \lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1
- \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a

Finding limits involving polynomials and other functions:
- \lim_{x \to 0} \frac{(1 + x)^n - 1}{x} = n