Evaluate the limit of 3x^3 + 5x^2 + 4x - 10 as x approaches 2.

Step-by-step solution

1. 1

   Check what kind of function you are taking the limit of. The expression $3x^3+5x^2+4x-10$ is a polynomial. Polynomials are built from sums and products of $x$ and constants, and every such function is continuous at every real number. Continuity at $a$ is exactly the statement

   $\lim_{x\to a}f(x)=f(a)$

   so no algebraic trickery is needed here — the whole problem reduces to an evaluation.

2. 2

   Substitute $x=2$ into each term separately. Working term by term keeps the arithmetic honest:

   $3(2)^3=3\cdot 8=24,\qquad 5(2)^2=5\cdot 4=20,\qquad 4(2)=8,\qquad -10$

3. 3

   Add the four pieces.

   $24+20+8-10=42$

   So

   $\lim_{x\to 2}\left(3x^3+5x^2+4x-10\right)=42$

4. 4

   Confirm that the substitution was legal. Direct substitution fails only when it produces an undefined form such as $\tfrac00$ or $\tfrac{c}{0}$. Here it produced the finite number $42$, so there is nothing indeterminate to resolve — no factoring, no conjugates, no L'Hôpital.

5. 5

   Sanity-check from both sides. Evaluating the polynomial slightly off $x=2$ gives $f(1.99)\approx 41.4023$ and $f(2.01)\approx 42.6023$; the values close in on $42$ from below and above, which is what a two-sided limit of $42$ looks like.