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- Lesson: 1e2:30

Limit is an important instrument that helps us understand ideas in the realm of Calculus. In this section, we will learn how to find the limit of a function graphically using one-sided limits and two-sided limits.

DEFINITION:

left-hand limit: $\lim_{x \to a^-} f(x) = L$ We say “the limit of f(x), as x approaches a from the negative direction, equals L”. It means that the value of f(x) becomes closer and closer to L as x approaches a from the left, but x is not equal to a.

DEFINITION:

right-hand limit: $\lim_{x \to a^+} f(x) = L$ We say “the limit of f(x), as x approaches a from the positive direction, equals L”. It means that the value of f(x) becomes closer and closer to L as x approaches a from the right, but x is not equal to a.

DEFINITION:

$\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = L$ and $\lim_{x \to a^-} f(x) = L$

left-hand limit: $\lim_{x \to a^-} f(x) = L$ We say “the limit of f(x), as x approaches a from the negative direction, equals L”. It means that the value of f(x) becomes closer and closer to L as x approaches a from the left, but x is not equal to a.

DEFINITION:

right-hand limit: $\lim_{x \to a^+} f(x) = L$ We say “the limit of f(x), as x approaches a from the positive direction, equals L”. It means that the value of f(x) becomes closer and closer to L as x approaches a from the right, but x is not equal to a.

DEFINITION:

$\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = L$ and $\lim_{x \to a^-} f(x) = L$

- 1.For the function f whose graph is shown, state the following:

a)$\lim_{x \to -5^-} f(x)$

$\lim_{x \to -5^+} f(x)$

$\lim_{x \to -5} f(x)$

$f(-5)$b)$\lim_{x \to -2^-} f(x)$

$\lim_{x \to -2^+} f(x)$

$\lim_{x \to -2} f(x)$

$f(-2)$c)$\lim_{x \to 1^-} f(x)$

$\lim_{x \to 1^+} f(x)$

$\lim_{x \to 1} f(x)$

$f(1)$d)$\lim_{x \to 4^-} f(x)$

$\lim_{x \to 4^+} f(x)$

$\lim_{x \to 4} f(x)$

$f(4)$e)$\lim_{x \to 5^-} f(x)$

$\lim_{x \to 5^+} f(x)$

$\lim_{x \to 5} f(x)$

$f(5)$

1.

Limits & Continuity

1.1

Finding limits from graphs

1.2

Limit laws

1.3

Finding limits algebraically - direct substitution

1.4

Finding limits algebraically - when direct substitution is not possible

1.5

Squeeze theorem

1.6

l'Hospital's rule

1.7

Infinite limits - vertical asymptotes

1.8

Continuity

1.9

Intermediate value theorem

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