Find the limit of the following functions as x approaches c: i) f(x) = x2, ii) f(x) = 1/x, iii) f(x)…

Question

Find the limit of the following functions as x approaches c: i) f(x) = x2, ii) f(x) = 1/x, iii) f(x) = sqrt(x)

ScanSolve AI Tutor · Accepted Answer

Here are the limits for each function: i) For f(x) = x²: Step 1: Substitute f(x) and f(c) into the limit expression. lim_(x → c) (x² − c²)/(x−c) Step 2: Factor the numerator using the difference of squares formula, a² − b² = (a−b)(a+b). lim_(x → c) ((x−c)(x+c))/(x−c) Step 3: Cancel out the (x−c) term, since x ≠ c as x → c. lim_(x → c) (x+c) Step 4: Substitute x=c into the simplified expression. c+c = 2c The limit is 2c. ii) For f(x) = 1/x: Step 1: Substitute f(x) and f(c) into the limit expression. lim_(x → c) (1/x − 1/c)/(x−c) Step 2: Combine the fractions in the numerator. lim_(x → c) ((c−x)/(xc))/(x−c) Step 3: Rewrite the expression by multiplying the numerator by the reciprocal of the denominator. lim_(x → c) (c−x)/(xc(x−c)) Step 4: Factor out −1 from the numerator to match the term in the denominator. lim_(x → c) (−(x−c))/(xc(x−c)) Step 5: Cancel out the (x−c) term. lim_(x → c) (−1)/(xc) Step 6: Substitute x=c into the simplified expression. (−1)/(c · c) = (−1)/(c²) The limit is −(1)/(c²). iii) For f(x) = √(x): Step 1: Substitute f(x) and f(c) into the limit expression. lim_(x → c) (√(x) − √(c))/(x−c) Step 2: Multiply the numerator and denominator by the conjugate of the numerator, √(x)+√(c). lim_(x → c) (√(x) − √(c))/(x−c) · (√(x)+√(c))/(√(x)+√(c)) Step 3: Apply the difference of squares formula (a−b)(a+b) = a² − b² to the numerator. lim_(x → c) ((√(x))² − (√(c))²)/((x−c)(√(x)+√(c))) lim_(x → c) (x−c)/((x−c)(√(x)+√(c))) Step 4: Cancel out the (x−c) term. lim_(x → c) (1)/(√(x)+√(c)) Step 5: Substitute x=c into the simplified expression. (1)/(√(c)+√(c)) = (1)/(2√(c)) The limit is (1)/(2√(c)).

Find the limit of the following functions as x approaches c: i) f(x) = x2, ii) f(x) = 1/x, iii) f(x) = sqrt(x)

Answer

Step-by-step solution

Part i· 4 steps

Substitute $f(x)$ and $f(c)$ into the limit expression.

Factor the numerator using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ .

Cancel out the $(x-c)$ term, since $x \neq c$ as $x \to c$ .

Substitute $x=c$ into the simplified expression.

Part ii· 6 steps

Substitute $f(x)$ and $f(c)$ into the limit expression.

Combine the fractions in the numerator.

Rewrite the expression by multiplying the numerator by the reciprocal of the denominator.

Factor out $-1$ from the numerator to match the term in the denominator.

Cancel out the $(x-c)$ term.

Substitute $x=c$ into the simplified expression.

Part iii· 5 steps

Substitute $f(x)$ and $f(c)$ into the limit expression.

Multiply the numerator and denominator by the conjugate of the numerator, $\sqrt{x}+\sqrt{c}$ .

Apply the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ to the numerator.

Cancel out the $(x-c)$ term.

Substitute $x=c$ into the simplified expression.

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Find the limit of the following functions as x approaches c: i) f(x) = x2, ii) f(x) = 1/x, iii) f(x) = sqrt(x)

Answer

Step-by-step solution

Part i· 4 steps

Substitute f(x)f(x)f(x) and f(c)f(c)f(c) into the limit expression.

Factor the numerator using the difference of squares formula, a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b)a2−b2=(a−b)(a+b).

Cancel out the (x−c)(x-c)(x−c) term, since x≠cx \neq cx=c as x→cx \to cx→c.

Substitute x=cx=cx=c into the simplified expression.

Part ii· 6 steps

Substitute f(x)f(x)f(x) and f(c)f(c)f(c) into the limit expression.

Combine the fractions in the numerator.

Rewrite the expression by multiplying the numerator by the reciprocal of the denominator.

Factor out −1-1−1 from the numerator to match the term in the denominator.

Cancel out the (x−c)(x-c)(x−c) term.

Substitute x=cx=cx=c into the simplified expression.

Part iii· 5 steps

Substitute f(x)f(x)f(x) and f(c)f(c)f(c) into the limit expression.

Multiply the numerator and denominator by the conjugate of the numerator, x+c\sqrt{x}+\sqrt{c}x​+c​.

Apply the difference of squares formula (a−b)(a+b)=a2−b2(a-b)(a+b) = a^2 - b^2(a−b)(a+b)=a2−b2 to the numerator.

Cancel out the (x−c)(x-c)(x−c) term.

Substitute x=cx=cx=c into the simplified expression.

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More Mathematics Questions

Substitute $f(x)$ and $f(c)$ into the limit expression.

Factor the numerator using the difference of squares formula, $a^2 - b^2 = (a-b)(a+b)$ .

Cancel out the $(x-c)$ term, since $x \neq c$ as $x \to c$ .

Substitute $x=c$ into the simplified expression.

Substitute $f(x)$ and $f(c)$ into the limit expression.

Factor out $-1$ from the numerator to match the term in the denominator.

Cancel out the $(x-c)$ term.

Substitute $x=c$ into the simplified expression.

Substitute $f(x)$ and $f(c)$ into the limit expression.

Multiply the numerator and denominator by the conjugate of the numerator, $\sqrt{x}+\sqrt{c}$ .

Apply the difference of squares formula $(a-b)(a+b) = a^2 - b^2$ to the numerator.

Cancel out the $(x-c)$ term.

Substitute $x=c$ into the simplified expression.