Sarah was new to Maplewood Elementary School and felt anxious about making new friends. She had recently moved to the town and everything felt unfamiliar. One sunny afternoon, during recess, she noticed a group of students gathered around a boy named Alex. They were teasing him, calling him hurtful names, and mocking his clothes. Alex looked very upset and helpless.
Max brought his new red truck to the park. He wouldn't let anyone play with it. "My truck!" he said. Mia asked nicely, but Max shook his head.
Then... OH NO! The truck got stuck in a tree! 🎄 Mia helped with her jump rope. Max smiled and said, "Let's play together!" They built an awesome ramp and shared the truck all afternoon. 🌞
Geometric: Directed line segment \( \overrightarrow{AB} \)
Algebraic: \( \mathbf{v} = \langle v_x, v_y, v_z \rangle \)
From \( A(1,2) \) to \( B(4,6) \): \( \mathbf{v} = \langle 3,4 \rangle \)
Transformations are split into two groups:
\( f(2x) \) compresses horizontally by \( \frac{1}{2} \)
\( f(-x) \) reflects over \( y \)-axis
\( f(x - 3) \) shifts right 3 units
Understanding limits is a fundamental concept in calculus. Evaluating left-hand limits (\( \lim_{x \to c^-} \)) and right-hand limits (\( \lim_{x \to c^+} \)) is essential to determine if a limit exists at a given point. This guide walks you through the steps to evaluate these limits and explains when they are equal or unequal, supported by examples to solidify your understanding.
Identify the function \(f(x)\) and the value \(c\) at which you want to evaluate the limit. Determine if \(f(x)\) is continuous, piecewise-defined, or has discontinuities near \(c\).
Approach the value of \(c\) from the left (values less than \(c\)). Substitute \(x\)-values slightly less than \(c\) into \(f(x)\) and observe the behavior.
Approach \(c\) from the right (values greater than \(c\)). Substitute \(x\)-values slightly greater than \(c\) into \(f(x)\) and observe the behavior.
If \( \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) \), the limit exists, and \(\lim_{x \to c} f(x)\) equals this common value. If \( \lim_{x \to c^-} f(x) \neq \lim_{x \to c^+} f(x) \), the limit does not exist.
Rule: \( \lim_{{x \to c}} k = k \)
Example:
\( \lim_{{x \to 4}} 7 = 7 \)
Rule: \( \lim_{{x \to c}} x = c \)
Example:
\( \lim_{{x \to 5}} x = 5 \)
Comprehensive guide to Limits in Calculus, designed for students in the MCV4U course.
A limit describes the value that a function f(x) approaches as the input x approaches a particular value. Limits are crucial for understanding continuity, derivatives, and integrals in calculus.
Notation:
\[ \lim_{x \to c} f(x) = L \]
This means as \(x\) approaches \(c\), \(f(x)\) approaches \(L\).
If \(f(x)\) approaches a specific value \(L\) as \(x \to c\), the limit exists.