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Mathematics for Data Science

bookIntroduction to Limits

Limits are a fundamental concept in calculus that describe how a function behaves as it approaches a specific point. They help define derivatives and integrals, forming the backbone of mathematical analysis and machine learning optimization techniques.

Formal Definition & Notation

A limit represents the value that a function approaches as the input gets arbitrarily close to a point.

lim⁡x→af(x)=L\lim_{x \rarr a}f(x) = L

This means that as xx gets arbitrarily close to aa where approaches LL.

Note
Note

The function does not need to be defined at x=ax=a for the limit to exist.

One-Sided & Two-Sided Limits

A limit can be approached from either side:

  • Left-hand limit: approaching aa from values smaller than aa. lim⁡x→a−f(x)\lim_{x \rarr a^-}f(x)
  • Right-hand limit: approaching aa from values larger than aa. lim⁡x→a+f(x)\lim_{x \rarr a^+}f(x)
  • The limit exists only if both one-sided limits are equal: lim⁡x→a−f(x)=lim⁡x→a+f(x)\lim_{x \rarr a^-}f(x) = \lim_{x \rarr a^+}f(x)

Evaluating Limits in Python

Python, through the sympy library, allows us to compute limits symbolically.

Computing a Limit Directly


              
12345678

            
import sympy as sp

x = sp.symbols('x')
f = (x**2 - 4) / (x - 2)

# Compute the limit as x approaches 2
limit_value = sp.limit(f, x, 2)
print("Limit of f(x) as x approaches 2:", limit_value)
copy

When Limits Fail to Exist

A limit does not exist in the following cases:

  • Jump discontinuity: lim⁡x→a−f(x)≠lim⁡x→a+f(x)\lim_{x \rarr a^-}f(x) \neq \lim_{x \rarr a^+}f(x)
    • Example: a step function where the left and right limits are different.
  • Infinite limit: lim⁡x→01x2=∞\lim_{x \rarr 0}\frac{1}{x^2}=\infty
    • The function grows unbounded.
  • Oscillation: lim⁡x→0sin⁡(1x)\lim_{x \rarr 0}\sin\left(\frac{1}{x}\right)
    • The function fluctuates infinitely without settling to a single value.

Evaluating a Limit That Does Not Exist


              
12345678

            
import sympy as sp

x = sp.symbols('x')
f = (x**2 - 4) / (x - 2)

# Compute the limit as x approaches 2
limit_value = sp.limit(f, x, 2)
print("Limit of f(x) as x approaches 2:", limit_value)
copy

Special Case – Limits at Infinity

When xx approaches infinity, we analyze the end behavior of functions:

  • Rational functions: lim⁡x→∞1x=0\lim_{x \rarr \infty}\frac{1}{x}=0
  • Polynomial growth: lim⁡x→∞x2x=∞\lim_{x \rarr \infty}\frac{x^2}{x}=\infty
  • Dominant term rule: lim⁡x→∞axmbxn={0, if m<n,ab, if m=n,±∞, if m>n.\lim_{x \to \infty} \frac{a x^m}{b x^n} = \begin{cases} 0,\ \text{if } m < n,\\ \frac{a}{b},\ \text{if } m = n, \\ \pm \infty,\ \text{if } m > n. \end{cases}

Computing a Limit at Infinity


              
1234567

            
import sympy as sp

x = sp.symbols('x')
f = 1 / x

limit_value = sp.limit(f, x, sp.oo)
print("Limit of 1/x as x approaches infinity:", limit_value)
copy

1. Which Python library is used to compute limits symbolically?

2. What will the following Python code output?

question mark

Which Python library is used to compute limits symbolically?

Select the correct answer

question mark

What will the following Python code output?

Select the correct answer

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bookIntroduction to Limits

Pyyhkäise näyttääksesi valikon

Limits are a fundamental concept in calculus that describe how a function behaves as it approaches a specific point. They help define derivatives and integrals, forming the backbone of mathematical analysis and machine learning optimization techniques.

Formal Definition & Notation

A limit represents the value that a function approaches as the input gets arbitrarily close to a point.

lim⁡x→af(x)=L\lim_{x \rarr a}f(x) = L

This means that as xx gets arbitrarily close to aa where approaches LL.

Note
Note

The function does not need to be defined at x=ax=a for the limit to exist.

One-Sided & Two-Sided Limits

A limit can be approached from either side:

  • Left-hand limit: approaching aa from values smaller than aa. lim⁡x→a−f(x)\lim_{x \rarr a^-}f(x)
  • Right-hand limit: approaching aa from values larger than aa. lim⁡x→a+f(x)\lim_{x \rarr a^+}f(x)
  • The limit exists only if both one-sided limits are equal: lim⁡x→a−f(x)=lim⁡x→a+f(x)\lim_{x \rarr a^-}f(x) = \lim_{x \rarr a^+}f(x)

Evaluating Limits in Python

Python, through the sympy library, allows us to compute limits symbolically.

Computing a Limit Directly


              
12345678

            
import sympy as sp

x = sp.symbols('x')
f = (x**2 - 4) / (x - 2)

# Compute the limit as x approaches 2
limit_value = sp.limit(f, x, 2)
print("Limit of f(x) as x approaches 2:", limit_value)
copy

When Limits Fail to Exist

A limit does not exist in the following cases:

  • Jump discontinuity: lim⁡x→a−f(x)≠lim⁡x→a+f(x)\lim_{x \rarr a^-}f(x) \neq \lim_{x \rarr a^+}f(x)
    • Example: a step function where the left and right limits are different.
  • Infinite limit: lim⁡x→01x2=∞\lim_{x \rarr 0}\frac{1}{x^2}=\infty
    • The function grows unbounded.
  • Oscillation: lim⁡x→0sin⁡(1x)\lim_{x \rarr 0}\sin\left(\frac{1}{x}\right)
    • The function fluctuates infinitely without settling to a single value.

Evaluating a Limit That Does Not Exist


              
12345678

            
import sympy as sp

x = sp.symbols('x')
f = (x**2 - 4) / (x - 2)

# Compute the limit as x approaches 2
limit_value = sp.limit(f, x, 2)
print("Limit of f(x) as x approaches 2:", limit_value)
copy

Special Case – Limits at Infinity

When xx approaches infinity, we analyze the end behavior of functions:

  • Rational functions: lim⁡x→∞1x=0\lim_{x \rarr \infty}\frac{1}{x}=0
  • Polynomial growth: lim⁡x→∞x2x=∞\lim_{x \rarr \infty}\frac{x^2}{x}=\infty
  • Dominant term rule: lim⁡x→∞axmbxn={0, if m<n,ab, if m=n,±∞, if m>n.\lim_{x \to \infty} \frac{a x^m}{b x^n} = \begin{cases} 0,\ \text{if } m < n,\\ \frac{a}{b},\ \text{if } m = n, \\ \pm \infty,\ \text{if } m > n. \end{cases}

Computing a Limit at Infinity


              
1234567

            
import sympy as sp

x = sp.symbols('x')
f = 1 / x

limit_value = sp.limit(f, x, sp.oo)
print("Limit of 1/x as x approaches infinity:", limit_value)
copy

1. Which Python library is used to compute limits symbolically?

2. What will the following Python code output?

question mark

Which Python library is used to compute limits symbolically?

Select the correct answer

question mark

What will the following Python code output?

Select the correct answer

Oliko kaikki selvää?

Miten voimme parantaa sitä?

Kiitos palautteestasi!

Osio 3. Luku 1
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