CHAPTER 2 – LIMITS AND CONTINUITY
2.1 Limits
Definition
limx→af(x)=L\lim_{x \to a} f(x) = Lx→alimf(x)=L
means as x approaches a, f(x) approaches L.
2.2 Standard Limits
Trigonometric Limits
limx→0sinxx=1\lim_{x\to 0}\frac{\sin x}{x} = 1x→0limxsinx=1
limx→01−cosxx2=12\lim_{x\to 0}\frac{1-\cos x}{x^2} = \frac{1}
{2}x→0limx21−cosx=21
Exponential Limits
limx→∞(1+1x)x=e\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x =
ex→∞lim(1+x1)x=e
Algebraic Limits
limx→0x∣x∣=Does not exist\lim_{x\to 0}\frac{x}{|x|} = \text{Does not
exist}x→0lim∣x∣x=Does not exist
2.3 Limit Evaluation Techniques
(a) Direct Substitution
If no indeterminate form exists.
(b) Factorization
Example:
2.1 Limits
Definition
limx→af(x)=L\lim_{x \to a} f(x) = Lx→alimf(x)=L
means as x approaches a, f(x) approaches L.
2.2 Standard Limits
Trigonometric Limits
limx→0sinxx=1\lim_{x\to 0}\frac{\sin x}{x} = 1x→0limxsinx=1
limx→01−cosxx2=12\lim_{x\to 0}\frac{1-\cos x}{x^2} = \frac{1}
{2}x→0limx21−cosx=21
Exponential Limits
limx→∞(1+1x)x=e\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^x =
ex→∞lim(1+x1)x=e
Algebraic Limits
limx→0x∣x∣=Does not exist\lim_{x\to 0}\frac{x}{|x|} = \text{Does not
exist}x→0lim∣x∣x=Does not exist
2.3 Limit Evaluation Techniques
(a) Direct Substitution
If no indeterminate form exists.
(b) Factorization
Example:
