Calculus

Limit and Continuity

Limit

A function f(x) has limit L as x approaches a if the limit from the left exists and the limit from right exists and both limit are L.

\(i.e lim_{x \to a^+} f(x) = \lim_{x \to a^-} f(x) = \lim_{x \to a} f(x)\)

Some important functions:

1. \(lim_{x \to 0} \frac{sinx}{x} = lim_{x \to 0} \frac{x}{sinx} = lim_{x \to 0} \frac{sin^{-1}x}{x} = lim_{x \to 0} \frac{x}{sin^{-1}x} = 1\)
2. \(\lim_{x \to ∞} \left ( 1 + \frac{1}{x} \right )^x = \lim_{x \to ∞} (1 + x)^{\frac{1}{x}} = e\)
3. \(\lim_{x \to 0} =\frac{tanx}{x} = 1\)
4. \(\lim_{x \to 0} = \frac{a^x – 1}{x} = loga\)
5. \(\lim_{x \to ∞} = \frac{logx}{x} = 0\)
6. \(\lim_{x \to 0} = \frac{log(1+x)}{x} = 1\)
7. When functional value is in the form \(\frac{0}{0}\),\(\frac{∞}{∞}\), ∞, -∞, ∞⁰, 0^∞, 1^∞ etc, it is said to be indeterminate form.
8. A function f(x) is said to be in infinitesimal as x → a if \(\lim_{x \to a} f(x) = 0\) (i.e limit of function is zero)
9. If \(\lim_{x \to a} f(x) = x\) and \lim_{x \to a} g(x) = y. If x and y are finite then \(\lim_{x \to a} f(x) g(x) = xy\)
   • \(\lim_{x \to a} f(x) \pm g(x) = x \pm y\)
   • \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{x}{y}\)
   • \(\lim_{x \to a} k f(x) = k\lim_{x \to a} f(x) = kx\)
   • \(\lim_{x \to a} [f(x) + k] = \lim_{x \to a} f(x) + k\)

L-Hospital Rule

Let f(x) and g(x) be two function such that f(a) = 0 and g(a) = 0

\(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f^{‘}(x)}{g^{‘}(x)} = \frac{f^{‘}(x)}{g^{‘}(x)}\)

provided that g^‘(a) ≠ 0

Note: This rule is applicable for \(\frac{0}{0}\) and \(\frac{∞}{∞}\) form.

Continuity

A function is said to be continuous at point x = a if the limiting value of the function is equal to functional value at the same point.

i.e.

Right Hand Limit = Left Hand Limit = Functional Value

i.e. \(\lim_{x \to a} f(x) = f(a)\)

i.e. f(a⁺) = f(a^–) = f(a)

f(x) is said to be continuous at x = a if \(\lim_{x \to a} f(a + h) = \lim_{x \to a} f(a – h) = f(a)\)

Note: A function f(x) defined in the neighborhood of the point x = a is said to be continuous at x = a if

\(\lim_{x \to a} f(x) = f(a)\) provided that

1. f(a) is defined
2. \(\lim_{x \to a} f(x) \) exists

Derivative and its applications

Derivative:

Let y = f(x) be a continuous function defined in the interval (a, b). The derivative or differential coefficient of f(x) with respect to x is defined by

\(\lim_{x \to 0} \frac{Δy}{Δx} = \lim_{x \to 0} \frac{f(x + Δx) – f(X)}{Δx}\)

It is denoted by \(\frac{dy}{dx}\) or f^‘(x) or y’ or y₁ or Df(x)

\(\frac{dy}{dx} = \lim_{Δx \to x} \frac{Δy}{Δx} = \lim_{Δx \to x}\frac{(x + Δx) – f(x)}{Δx}\)

Note: Every differentiable function is continuous, but not conversely.