Q. 0C - Calculus

Question

Read the section and make your own summary of the material.

Step-by-Step Solution

Answer

The summary is explained in Step 2.

Step 1. To find

The summary of the section \(1.5\)

Step 2. Explanation

• Constant multiple rule: \(\lim_{x\rightarrow c}k.f(x)=k.\lim_{x\rightarrow c}f(x)\)
  Sum rule: \(\lim_{x\rightarrow c}(f(x)+g(x))=\lim_{x\rightarrow c}f(x)+\lim_{x\rightarrow c}g(x)\)
  Difference rule: \(\lim_{x\rightarrow c}(f(x)-g(x))=\lim_{x\rightarrow c}f(x)-\lim_{x\rightarrow c}g(x)\)
  Product rule: \(\lim_{x\rightarrow c}(f(x)\cdot g(x))=\lim_{x\rightarrow c}f(x)\cdot \lim_{x\rightarrow c}g(x)\)
  Quotient rule: \(\lim_{x\rightarrow c}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow c}f(x)}{\lim_{x\rightarrow c}g(x)}\) if \(\lim_{x\rightarrow c}g(x)\neq 0\)
  Composition rule: \(\lim_{x\rightarrow c}(f(g)(x))=f(\lim_{x\rightarrow c}g(x))\), if f is continuous at \(\lim_{x\rightarrow c}g(x)\)
• If \(f(x)\) and \(f(x)\) continuos at \(x=c\) and 'k' is any constant, then the functions \(k.f(x),f(x)\pm g(x),f(x).g(x)\) are also continuous  at \(x=c\). Moreover, if \(g(c)\neq 0\), then \(\frac{f(x)}{g(x)}\) is continuous at \(x=c\), and if \(f(x)\) is also continuous at \(g(c)\) then \(f(g(x))\) is continuous at \(x=c\).
• All algebraic functions are continuous on their domains.
• If \(\lim_{x\rightarrow c} g(x)\) exists, and \(f(x)\) is a function that is equal to \(g(x)\) for all \(x\) sufficiently close to \(c\) except possibly at \(c\) itself, then \(\lim_{x\rightarrow c}f(x)=\lim_{x\rightarrow c} g(x)\).
• If \(l(x)\leq f(x)\leq u(x)\) for all x sufficiently close to \(c\), but not neccessarily at \(x=c\), and if \(\lim_{x\rightarrow c} u(x)\) & \(\lim_{x\rightarrow c} l(x)\) are both equal to L, then \(\lim_{x\rightarrow c} f(x)=L\).
• All exponential and logarithmic functions are continuous on their domains.
• All trigonometric and inverse trigonometric functions are continuous on their domains.
• \(\lim_{h\rightarrow 0}(1+h)^{\frac{1}{h}}=e\)
• \(\lim_{h\rightarrow 0}{\frac{e^{h}-1}{h}}=1\)